Endodontic files with cross-cuts

ABSTRACT

Endodontic instruments are described which have at least a section with a center of mass offset from an axis of rotation so that when the instrument is rotated, the section bends away from the axis of rotation.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.14/632,930, filed Feb. 26, 2015, which claims priority to and is acontinuation of U.S. application Ser. No. 14/455,636, filed Aug. 8,2014, which claims priority to and is a continuation of U.S. applicationSer. No. 13/804,084 (issued as U.S. Pat. No. 8,882,504), filed Mar. 14,2013, which claims priority to and is a continuation of U.S. applicationSer. No. 11/402,207 (issued as U.S. Pat. No. 8,454,361), filed Apr. 10,2006, which claims priority to U.S. Provisional Application Ser. No.60/669,409, filed Apr. 8, 2005. The disclosure of the prior applicationsis considered part of and is incorporated by reference in the disclosureof this application.

BACKGROUND

The present invention relates to endodontic instruments.

Endodontic instruments can be used for cleaning and enlarging theendodontic cavity space (“ECS”), also known as the root canal system ofa human tooth. FIG. 1A shows an example of an unprepared root canal 102of a tooth 104. As can be seen, the unprepared root canal 102 is usuallya narrow channel that runs through the central portion of the root ofthe tooth. Cleaning and enlargement of the ECS can be necessitated bythe death or necrosis of the dental pulp, which is the tissue thatoccupies that space in a healthy tooth. This tissue can degenerate for amultitude of reasons, which include tooth decay, deep dentalrestorations, complete and incomplete dental fractures, traumaticinjuries or spontaneous necrosis due to the calcification and ischemiaof the tissue, which usually accompanies the ageing process. Similar toa necrotic or gangrenous appendix, the complete removal of this tissueis paramount, if not urgent, because of the subsequent development ofinfections or dental abscesses, septicemia, and even death.

The root canal system of a human tooth is often narrow, curved andcalcified, and can be extremely difficult to negotiate or clean. Indeed,the conventional endodontic or root canal instruments currentlyavailable are frequently inadequate in the complete removal of the pulpand the efficient enlargement of the ECS. Furthermore, they are usuallypredisposed to breakage, causing further destruction to the tooth.Broken instruments are usually difficult, if not impossible to remove,often necessitating the removal of the tooth. Injury to the tooth, whichoccurs as the result of a frank perforation or alteration of the naturalanatomy of the ECS, can also lead to failure of the root canal and toothloss.

A root canal procedure itself can be better appreciated by referring toFIGS. 1A and 1B. The unprepared root canal 102 of the tooth 104 usuallybegins as a narrow and relatively parallel channel. The portal of entryor the orifice 106 and the portal of exit or foramen 108 are relativelyequal in diameter. To accommodate complete cleaning and filling of thecanal and to prevent further infection, the canal must usually beprepared. The endodontic cavity preparation (“ECP”) generally includesprogressively enlarging the orifice and the body of the canal, whileleaving the foramen relatively small. The result is usually a continuouscone shaped preparation, for example, the space 109.

In general, endodontic instruments are used to prepare the endodonticcavity space as described above. Endodontic instruments can include handinstruments and engine driven instruments. The latter can but need notbe a rotary instrument. Combinations of both conventional hand andengine driven rotary instruments are usually required to perform an ECPsuccessfully and safely.

FIGS. 2A and 2B show a conventional endodontic instrument 200. Theendodontic instrument shown includes a shaft 202 that includes a tip 204and a shank 206. The endodontic instrument 200 also includes grooves 208and 210 that spiral around the shaft 202. The grooves are referred to inthe instant specification as flutes.

FIG. 2B shows a cross section 212 (i.e., cross section A-A) of theendodontic instrument. The cross section 208 shows cross sections 214and 216 of flutes 208 and 210, respectively. As can be seen from FIGS.2A and 2B, the flutes 208 and 210 are generally the spacing on bothsides of a helical structure 218 (or helix) that spirals around theshaft 202. The bottom portion of a flute—seen as a line or curve (e.g.,curve 220 indicated in bold)—is referred to in the instant specificationas a spline (indicated by line in bold). The portion of a spline thatcomes into contact with a surface being cut during cutting will bereferred to in the instant specification as a radial land. Item 222 ofFIG. 2B is an example of a radial land.

A flute of an endodontic instrument usually includes a sharpened edgeconfigured for cutting. Edge 224 of FIG. 2A is an example of such acutting edge. Edge 224 can be seen as a point 226 in FIG. 2B. Generally,an instrument having right handed cutting edges is one that will cut orremove material when rotated clockwise, as viewed from shank to tip. Inthis specification, a direction of rotation will be specified as viewedfrom the shank to the tip of the instrument. The cut direction ofrotation for a right handed endodontic instrument is clockwise. Aninstrument having left handed cutting edges is one that will cut orremove material when rotated counter clockwise. The cut direction ofrotation, in this case, is counter clockwise.

An endodontic instrument includes a working portion, which is theportion that can cut or remove material. The working portion istypically the portion along the shaft that is between the tip of theinstrument and the shank end of the flutes. Portion 228 is the workingportion for the endodontic instrument shown in FIG. 2A. The workingportion is also referred to in this specification as the cuttingportion, and the working length as the cutting or working length.

Hand instruments are typically manufactured from metal wire blanks ofvarying sizes. The metallurgical properties of these wires, in general,have been engineered to produce a wide range of physical properties.These wires are usually then twisted or cut to produce specific shapesand styles. Examples of hand instruments include K-type, H-type, andR-type hand instruments. FIG. 2C show a barbed broach 230, which is oneexample of an R type instrument. FIG. 2D shows a cross section 232(i.e., cross section A-A) of the barbed broach 230. The barbed broach ismanufactured from soft iron wire that is tapered and notched to formbarbs or rasps along its surface. These instruments are generally usedin the gross removal of pulp tissue or debris from the root canalsystem. Another R-type file is a rat-tail file.

K-type instruments in current usage include reamers and K-files. FIG. 2Eshows an example of a K file 234. FIG. 2F shows a cross section 236(i.e., cross section A-A) of the K file 234. K files are generallyavailable in carbon steel, stainless steel, and more recently, an alloyof nickel-titanium. To fabricate a K type instrument, a round wire ofvarying diameters is usually grounded into three or four-sided pyramidalblanks and then rotated or twisted into the appropriate shapes. Theseshapes are specified and controlled by the American National StandardsInstitute (“ANSI”) and the International Standards Organization (“ISO”).The manufacturing processes for reamers and files are similar; excepthowever, files usually have a greater number of flutes per unit lengththan reamers. Reamers are used in a rotational direction only, whereasfiles can be used in a rotational or push-pull fashion. Files made fromthree-sided or triangular blanks have smaller cross sectional areas thanfiles made from four sided blanks. Thus, these instruments are usuallymore flexible and less likely to fracture. They also can display largerclearance angles and are more efficient during debridement. Triangularfiles, therefore, are generally considered more desirable for handinstrumentation.

FIG. 2G shows an example of an H-type file 238. FIG. 2H shows a crosssection 240 (i.e., cross section A-A) of the H type file 238. H typefiles are usually manufactured by grinding flutes into tapered roundmetal blanks to form a series of intersecting cones. H type files canusually cut only in the pull direction (i.e., a pull stroke). Primarilybecause they have positive rake angles, H type files can be extremelyefficient cutting instruments.

Hand instruments are usually manufactured according to guidelines of theANSI and the ISO, which specified that a working portion of aninstrument be 16 mm in length. ANSI and ISO further specified that afirst diameter or D1 of the instrument, be 1 mm from the tip or D₀.Other ANSI and ISO specifications require that: instruments have astandard taper of 0.02 mm per mm along the working portion 216; the tipmaintain a pyramidal shape no greater than 75° in linear cross section;and hand instruments (e.g., the ones shown in FIGS. 2A 2H) be availablein 21, 25, and 31 mm lengths.

In addition to the hand instruments described above, there are rotaryinstruments that are usually motor driven. FIG. 3A shows an examplerotary instrument 300 that is referred to as a G type reamer or drill.FIG. 3B shows a cross section 301 (i.e., cross section A-A) of the Gtype instrument. G-type drills are usually available in carbon orstainless steel. As is typical, the G type drill 300 shown includes ashort flame-shaped head 302 attached to a long shank 303. The core orweb shown in FIG. 3B shows the cross sections 304, 305, and 306 of threeflutes. The flutes, in this instance, have U shaped splines. Theinstrument 300 includes cutting edges that have negative rake-angles. Ingeneral, a rake angle is the angle between the leading edge of a cuttingtool and a perpendicular to the surface being cut. Rake angle is furtherdescribed below. The flame shaped head 302 includes a non cuttingsurface to prevent perforation. The instrument 300 is usually used as aside cutting instrument only. The instrument 300 is relatively rigidand, therefore, cannot usually be used in a curved space, for example,the ECS.

G-type drills are available in 14, 18 and 25 mm lengths as measured fromtip to shank, which is where the drill can be inserted into a standardslow-speed hand piece via a latch grip 307. G type drills are availablein varying diameters of 0.30 mm to 1.5 mm and from sizes 1 through 6.

SUMMARY

The present invention provides methods and apparatus for providingswaggering endodontic instruments for preparing an endodontic cavityspace.

In one aspect, the invention is directed to an endodontic device havinga tapered body having a tip end and a shank end, wherein the tip end hasa diameter that is less than a diameter of the shank end and the bodyhas an axis of rotation extending from the tip end to the shank end. Thebody has at least one working surface extending between the shank endand the tip end; a first cross section perpendicular to the axis ofrotation, wherein the first cross section has a first geometry; and asecond cross section perpendicular to the axis of rotation, and whereinthe second cross section has a second geometry, wherein the firstgeometry is different from the second geometry; wherein the first crosssection and the second cross section intersect the at least one workingsurface.

Implementations may include one or more of the following features. Thefirst geometry can be symmetrical and the second geometry can beasymmetrical. The first cross section can be closer to the shank endthan the tip end. The first cross section can have a different number ofworking surfaces than the second cross section. At the second crosssection, a centroid can be offset from the axis of rotation. The firstgeometry and the second geometry can include different numbers ofworking surfaces. The body can be flexible. The body can be formed ofnickel-titanium. The body can be sufficiently flexible such that when atip of the body is bound at a fixed position as the body rotates, aportion of the body that intersects the second cross section bends awayfrom the axis of rotation a substantially equal amount at a first angleof rotation and at a second angle of rotation. The first angle ofrotation can be 180° from the second angle of rotation. The second crosssection can bend away from the axis of rotation a substantially equallyamount at each angle of rotation. A non-swaggering portion of the bodycan have a centroid that lies substantially on the axis of rotation andintersects the at least one working surface as the tip of the body isbound at a fixed position and the body rotates. The at least one workingsurface can include a cutting flute. A tip of the body may not havecutting surfaces. At a cross section that intersects the axis ofrotation, a center of mass may be offset from the axis of rotation. Theworking surface can be configured to remove material when the body isrotated within a canal of the material. When the tip of the body is heldin place and the body is rotated, at least a portion of the body mayform helical waves

In another aspect, the invention is directed to an endodontic device,comprising a tapered body having a tip end and a shank end, wherein thetip end has a diameter that is less than a diameter of the shank end andthe body has an axis of rotation, wherein along the length of thetapered body, at least one centroid of a cross section that intersects aworking surface of the body is on the axis of rotation and at least onecentroid is offset from the axis of rotation, the body having at leastone cutting surface along the outer diameter that is configured toremove material when the body is rotated within a canal of the material.

Implementations may include one or more of the following features. Thebody can be canted so that the centroid on the axis of rotation is nearthe shank end and a tip is furthest from the axis of rotation ascompared to other centroids along the length of the body. The body cabbe linear. A plurality of centroids that correspond to the cuttingsurface may be in a line. The body can have a portion including thecutting surface and the portion can be linear. The body can be flexible.The body can be formed of nickel-titanium. The body can be sufficientlyflexible such that when a tip of the body is bound at a fixed positionas the body rotates, a portion of the body can bend away from the axisof rotation a substantially equal amount at multiple angles of rotation.The body can be curved. The tip can be offset from the axis of rotation.The tip can be on the axis of rotation. The body can have at least oneworking surface extending between the shank end and the tip end; a firstcross section perpendicular to the axis of rotation, wherein the firstcross section has a first geometry; a second cross section perpendicularto the axis of rotation, wherein the second cross section has a secondgeometry, wherein the first geometry is different from the secondgeometry; and the first cross section and the second cross sectionintersect the at least one cutting surface. When the tip of the body isheld in place and the body is rotated, at least a portion of the bodycan form helical waves.

In yet another aspect, the invention is directed to an endodonticdevice, comprising a tapered body having a tip end and a shank end,wherein the body has at least a portion with a center of mass that isoffset from an axis of rotation and is sufficiently flexible such thatwhen a tip of the body is bound at a fixed position as the body rotates,the portion bends away from the axis of rotation a substantially equalamount at a first angle of rotation and at a second angle of rotation.The first angle of rotation can be 180° from the second angle ofrotation. The portion can bend away from the axis of rotation asubstantially equal amount at each angle of rotation.

In another aspect, the invention is directed to endodontic device,comprising a linear body having a tip end and a shank end and a workingsurface between the tip end and the shank end, wherein a first crosssection of the body intersects the working surface toward the shank endand a second cross section of the body intersects the working surfacetoward the tip end, the first cross section is parallel to the secondcross section, both the first cross section and the second cross sectionare symmetrical and a first axis through a center of the first crosssection and perpendicular to the first cross section is different from asecond axis through a center of the second cross section andperpendicular to the second cross section.

Implementations may include one or more of the following features. Thebody can be flexible, e.g., the body can be formed of nickel-titanium.The body can be sufficiently flexible such that when a tip of the bodyis bound at a fixed position as the body rotates, a portion of the bodybends away from the axis of rotation a substantially equal amount atmultiple angles of rotation. The multiple angles of rotation can be 180°apart. The body can have the same geometry and dimensions down itslength. When the tip of the body is held in place and the body isrotated, at least a portion of the body can form helical waves.

In yet other aspect, the invention is directed to a method of cleaningan endodontic cavity space. A flexible instrument is inserted into theendodontic cavity space, wherein the instrument has a first portion witha center of mass that overlaps with an axis of rotation of theinstrument and at a second portion with a center of mass that is offsetfrom the axis of rotation. A tip of the flexible instrument is contactedagainst an inner surface of the cavity space. The instrument is rotatedso that the first portion bends away from the axis of rotation asubstantially equal amount at a first angle of rotation and at a secondangle of rotation. Rotating the instrument can include causing theinstrument for form sinusoidal waves within the cavity space.

In yet another aspect, the invention is directed to a method of cleaningan endodontic cavity space. A flexible instrument is inserted into theendodontic cavity space, wherein the instrument has a tapered bodyhaving a tip end and a shank end, the tip end has a diameter that isless than a diameter of the shank end and the body has an axis ofrotation extending from the tip end to the shank end, the body having atleast one working surface extending between the shank end and the tipend, a first cross section perpendicular to the axis of rotation,wherein the first cross section has a first geometry, and a second crosssection perpendicular to the axis of rotation, wherein the second crosssection has a second geometry, wherein the first geometry is differentfrom the second geometry, wherein the first cross section and the secondcross section intersect the at least one working surface. A tip of theflexible instrument is contacted against an inner surface of the cavityspace. The instrument is rotated so that the least one working surfaceintersecting the second cross section bends away from the axis ofrotation a substantially equal amount at a first angle of rotation andat a second angle of rotation.

Devices described herein can provide more efficient endodontic cleaningwhich is safer for a patient. An instrument that is both flexible andstrong resists breaking and injuring the patient. An instrument that isflexible and has a center of mass offset from an axis of rotation mayswing out from the axis of rotation as the instrument is rotated at highspeeds, such as when the instrument is used with a motorized tool. Ifthe instrument is configured to bend an equal amount at each angle ofrotation, the inner diameter of a space can be contacted by theinstrument and uniformly cleaned. The instrument can be made to have asmaller diameter than the space that requires cleaning, thereby allowingfor a difficult to access area to be accessed. The details of one ormore embodiments of the invention are set forth in the accompanyingdrawings and the description below. Other features, objects, andadvantages of the invention will be apparent from the description anddrawings, and from the claims.

DESCRIPTION OF DRAWINGS

FIGS. 1A and 1B illustrate a root canal procedure.

FIGS. 2A-2H show examples of endodontic instruments.

FIGS. 3A and 3B show an example of a rotary type endodontic instrument.

FIG. 4 shows an endodontic instrument having reversed helices.

FIG. 5 shows another endodontic instrument having reversed helices.

FIGS. 6A and 6B show other endodontic instruments having a reversedhelices.

FIGS. 7A-7D show an endodontic instrument having cross cuts on itshelices.

FIGS. 8A and 8B show an endodontic instrument having S splines, positiverake angles, and none or reduced radial lands.

FIGS. 9A-9H show an endodontic instrument having rolled edges andtapering.

FIGS. 10A-10C illustrate the notion of the critical set of endodonticinstruments.

FIGS. 11A-11D illustrate the contact area of an endodontic instrument.

FIG. 12 illustrates the working portion of a critical path set ofendodontic instruments.

FIG. 13 shows a path of a mass moving in a circle.

FIG. 14 shows a schematic of a portion of an instrument have atrapezoidal cross section.

FIGS. 15A-15B show the cross sections of the instrument in FIG. 14.

FIGS. 16A-16D show bodies with centers of mass different from theircenters of rotation.

FIG. 17 shows a stress versus strain diagram.

FIGS. 18A-18B show deflection of a cantilevered beam.

FIG. 19 shows a portion of an instrument.

FIG. 20A-20D show a beam with various axes illustrated.

FIGS. 21A-21E show one implementation of an endodontic instrument.

FIGS. 22A-22D show an endodontic instrument that includes a cutting tip.

FIGS. 23A-23D show another implementation of an endodontic instrument.

FIGS. 24A-24E show another implementation of an endodontic instrument.

FIGS. 25A-25E show another implementation of an endodontic instrument.

FIG. 26 illustrates swagger.

FIGS. 27A-30B show schematic representations of other implementations ofendodontic instruments.

FIGS. 31A-31B are a schematics of an swaggering instrument duringrotation.

FIGS. 32A-35B are schematics of blanks used for forming the instrumentsdescribed herein.

Like reference numbers and designations in the various drawings indicatelike elements.

DETAILED DESCRIPTION

Reversed Helix

Conventional endodontic instruments have right-handed cutting edges andright handed helices. A flute that forms a right handed helix spiralsaround the shaft, in a shank to tip longitudinal direction and,furthermore, in a clockwise direction of rotation (as viewed from shankto tip). This configuration is similar to the treads of a screw.Conventional endodontic instruments having this screw like configurationare prone to binding. Furthermore, the radial lands and negative cuttingangles typical of convention endodontic instruments predisposed theinstruments to premature fatigue and breakage.

An endodontic instrument in accordance with the invention can include areversed helix. A reversed helix spirals around the shaft of aninstrument, in an shank to tip longitudinal direction and, furthermore,in a direction of rotation opposite to the cut direction of rotation.If, for example, the endodontic instrument has a clockwise cut directionof rotation, its helices would spiral in a counter clockwise directionof rotation (along a longitudinal axis of the shaft in a shank to tipdirection and as viewed from shank to tip). In this case, the instrumentincludes right handed cutting edges. That is, the cutting edge is on theleading edge side of the helices as the instrument is rotated in the cutdirection of rotation.

FIG. 4 shows an example of the described endodontic instrument. Theinstrument shown includes a shaft 402, helices 404 and 406, shank end(or simply end) 408, tip end (or simply tip) 410. The helix 406, forexample, includes cutting edge 412. Thus, the instrument cuts when it isrotated about its longitudinal axis 418 in a counter clockwise direction(as seen from an end to tip perspective), and the cut direction iscounter clockwise (as indicated by arrow 414). The direction whichhelices 404 and 406 spiral around the shaft 402 is clockwise (along theshaft in an end to tip direction and as viewed from an end to tipperspective; as indicated by arrow 416).

If the endodontic instrument has a clockwise cut direction of rotation(as seen from an end to tip perspective), then its flutes can spiral ina counter clockwise direction of rotation (along a longitudinal axis ofthe shaft in an end to tip direction; and as viewed from an end to tipperspective). FIG. 5 shows and example of the described endodonticinstrument. The instrument shown includes a shaft 502, helices 504 and506, shank end (or simply end) 508, tip end (or simply tip) 510. Thehelix 504, for example, includes cutting edge 512. Thus, the instrumentcuts when it is rotated about its longitudinal axis 518 in a clockwisedirection (as seen from an end to tip perspective), and the cutdirection is clockwise (as indicated by arrow 514). The direction whichflutes 504 and 506 spiral around the shaft 502 is counter clockwise(along the shaft in an end to tip direction and as viewed from an end totip perspective; as indicated by arrow 516).

An endodontic instrument having the reversed helix is generally notprone to binding and can haul debris from its tip to its end, thusremoving the debris from the space being prepared. In oneimplementation, the endodontic instrument can be fabricated from aflexible or super flexible material, such as NiTi or a NiTi alloy.Engine driven instruments, including rotary engine driven instruments,as well as hand instruments can include the described reversed helixconfiguration.

In the above examples, the endodontic instruments shown included onlytwo flutes. Endodontic instruments having any number of flutes and anyspline geometry can incorporate the described reversed helixconfiguration. FIGS. 6A and 6B show implementations of multiple fluteendodontic instruments having the reversed helix configuration.

Helices Having Cross Cuts

An endodontic instrument in accordance with the invention can includehelices that include one or more cross cuts. The cross cuts of a helixcan but need not be at right angles to the helix. In general, the crosscuts can have a geometry and depth so as to increase the flexibility ofthe endodontic instrument and allow the instrument to bend more easily.FIG. 7A shows an instrument 702 that includes helices having cross cuts.FIG. 7B shows a cross section 704 (i.e., cross section B-B) of theinstrument 702. FIG. 7C shows the different example geometries which across cut can have. The cross cuts can include cutting edges, forexample, cutting edge 706, which consequently provide a more efficientcutting device. FIG. 7D shows a cross section 708 (i.e., cross sectionA-A) of the instrument 702.

Web Designs Having S Shaped Splines, Positive Rake Angles, and None orReduced Radial Lands

FIG. 8A shows an instrument 802, which is one example an endodonticinstrument having an S shaped spline, which approximate or approachpositive rake angles, and none or reduced radial lands. The instrument802 includes four helices 804, 806, 808, and 810 and four flutes 812,814, 816, and 818. FIG. 8B shows a cross section 820 (i.e., crosssection A-A) of the instrument 802. The web design shown exhibits aquadrilateral like shape. FIG. 8B shows cross sections 822, 824, 826,and 828 of the flutes. The splines are S shaped, which provides massthat can buttress the cutting edges of the instrument.

The cutting edges (shown as four arcs delimited by points 830, 832, 834,and 836) can have reduced positive rake angles, which makes the cuttingedges less prone to breakage than cuttings edge with large cuttingangles. In the instant specification, a cutting angle of a cutting edgethat is formed by a flute can be defined as the angle between (i) atangent of the spline of the flute at the cutting edge and (ii) a rayextending radially outward from the center of cross section of theinstrument For example, the cutting edge at point 836 that is formed byflute 822 exhibits a cutting angle 2 ₁ defined by tangent 840 and ray842. Tangent 840 can be mathematically represented as a one-sidedderivative, taken at point 836, of a function that represents the spline844. Alternatively, there are other ways of defining cutting angle. Forexample, the cutting angle can be defined as the angle 2 ₂ between thedescribed tangent 840 and a tangent 846 of a circumference 848 of theinstrument at point 836. Under the alternative definition, the cuttingangle 2 ₂ is said to be neutral or zero when the angle is ninetydegrees, positive when greater than ninety degrees, and negative whenless than ninety degrees.

An S shaped spline also removes the radial land usually present inconventional endodontic instruments. The crossed hatched area 850represents a hypothetical radial land. As can be seen, the radial land,if present, would rub against a surface being cut and create unnecessarydrag along the working surface of the instrument and render itinefficient and predisposed to breakage.

The described endodontic instrument can be fabricated from a preformedcylindrical metal blanks of nickel titanium. Alternatively, theinstrument can be fabricated from others blanks and other materials.

The shank end of the above described instrument can include a latch-typeattachment suitable for coupling, usually detachably, to a motor drivenchuck. The latch type attachment can also be suitable for coupling to ahandle if the instrument is to be used manually. The tip of theinstrument can be smooth while maintaining the conicity, taper, andtransverse cross sectional shape of the instrument.

The following describes an implementation. The tip of the implementationends in a pyramidal or parabolic shape and is at least 0.05 mm indiameter and 1-3 mm in length. The cutting length (not including thetip) of the implementation is 8-16 mm in length. In general, the cuttinglength should be at least 2 mm in length. The cutting edges of theimplementation is created by including one to six flutes. Alternatively,the implementation can include additional flutes. The flutes usuallybegin at a first position near the shank end of the instrument and endsat a second position near the tip end of the instrument. The firstposition is referred to as a maximum flute diameter (or MxFD) and thesecond position is referred to as a minimum flute diameter (or MnFD).The flutes are concave and are substantially the same as each other. Theflutes have a shape and depth that remains constant along the length ofthe shaft. Alternatively, the shape and/or depth can vary along thelength of the shaft. These flutes are spaced along the circumference ofthe cutting surface. The spacing can be of uniform intervals orirregular intervals. That is, the helix formation or spirals thatprogress from the shank end to the tip of the instrument can be spacedat regular intervals or increasingly narrower intervals. In the lattercase, a greater number of spirals can be included per unit length alongthe longitudinal axis of the implementation. Each flute forms a neutralor slightly positive rake angle. The flutes spiral around the shaft ofthe instrument, completing 360° of rotation for a minimum of 1 mm, and amaximum of 6 mm of axial length of the cutting surface.

Variable Working Surfaces with Flute Modifications and/or AttenuatedCutting Edges

The working surfaces or leading edges of conventional endodonticinstruments have been manufactured with active or sharp cutting edgesalong the entire length of the working surface. This configuration canpredispose the instrument to great amounts of torque leading topremature fatigue and breakage. One can mitigate the problem by varyingthe taper and the length of the working surface. As the instrumentincreased in diameter, the length of the working surface, or the numberof cutting flutes (i.e., the number of flutes that form cutting edges)per unit length along the longitudinal axis of the instrument, can bereduced. Although, this configuration does mitigate the amount of torquethat the instrument engenders as the size of the instrument increases,eliminating flutes also eliminates the ability of the instrument tocontinue to haul debris coronally. As a result, the instrument canbecome easily clogged creating unnecessary drag on the instrument.

An endodontic instrument in accordance with the invention retains theflutes along the entire length of the working surface to maintainhauling action. The leading edge or the working surface, however, ismodified such that only a portion of the working surface cuts. Thismodification is brought about by blunting or rolling the edge of flutesboth at the tip and shank ends of the instrument, leaving the centralportion of the cutting surface active. Rolling edges will prevent theinstrument from over-enlarging or tearing the foramen of the ECSdistally and mitigate drag and pre-mature fatigue proximally. FIGS. 9A9D show an example instrument that includes a reduced working portion.FIG. 9A shows the instrument 902. FIG. 9B shows the working portion 904of the instrument 902. FIG. 9C shows a non cutting tip portion 906 ofthe instrument 902. FIG. 9D shows a non cutting shank end portion 908 ofthe instrument 902. FIGS. 9E 9H show an implementation in which theinstrument 902 tapers from shank to tip.

Endodontic instruments can be provided in sets. A set usually includesinstruments of different diameters. In preparing an ECS, an endodontistusually begins the preparation process using the instrument having thesmallest diameter. As the ECS is enlarged, the endodontist usuallyswitches to instruments of progressively larger diameters. The rollededges feature described above can vary from one instrument to another ina set of instruments, with the active surface diminishing in lengthprogressively as the diameter of the instrument increased. This featurewould allow the instrument to continue to haul debris coronally, butmitigate the torque that the instrument is subject to when cutting.

Critical Path Set of Endodontic Instruments

A set of endodontic instruments, the instruments of which can have thesame length but different diameters, can be provided for use in seriatimto enlarge progressively the ECS 102 of a tooth (FIG. 1A) and create thecontinuous cone shaped space 109 (previously shown in FIG. 1B and alsoshown in FIG. 10A). In particular, the instrument with the smallestdiameter is first used to enlarge the ECS 102 to a certain extent. Then,the instrument with the next largest diameter is used to further enlargethe ECS. This process continues until the instrument having the largestdiameter is used to complete the creation of the space 109.Progressively enlarging the ECS 102 as described above can usuallysubject the tooth and the endodontic instruments to stress that issignificantly less than the stress that would be exerted if only oneinstrument were used to created the space 109.

The stress level to which an endodontic instrument is subject can berelated to the amount of material the instrument removes. In view of theforgoing, the number of endodontic instruments provided in the set aswell as their sizing can be determined so that each instrument of theset is subject to a similar or the same level of stress during its useto remove its apportioned amount of dental material. The determinationcan be based on a critical set model that will be explain in thefollowing paragraphs.

The indicated portion of the space 109 (FIG. 10A), which generallystarts at the point in the ECS (referred to below as the curve startpoint) where a straight line preparation is no longer practical orpossible and ends at the foramen of the ECS, can be modeled as thegeometric object 1002 shown in FIG. 10B. The object 1002 represents thecurved portion of an ideally prepared ECS.

The object 1002 can be incremented along its length, which is typically12 millimeters, into 12 one millimeter sections. As can be seen, thetransverse cross sections that so increment the space 109 are circlesthat are denoted as 50, 51, . . . , 62, where the cross section 50 is atthe tip and cross section 62 is at the butt of the space 109.

Note that the shaft (not shown) of each endodontic instrument of the setcan be similarly incremented by transverse cross sections denoted as D₁,D₂, . . . , D₁₂, each of which can correspond to a respective crosssection of the space 109. D₆, for example, corresponds to cross section56.

The cross section 56 represents the portion of the ECS that has thegreatest curvature, usually referred to as the fulcrum. The fulcrumusually occurs at or near the midpoint between the foramen and the abovedescribed curve point of the ECS and, furthermore, is generally thepoint where endodontic instruments are most vulnerable to failure,including, for example, catastrophic breakage.

The set of endodontic instruments can be configured to reduce theprobability of failure at or near the D₆ cross sections of their shafts.In particular, the set can be configured so that the total amount ofdental material to be removed at and around cross section 56 is evenlydistributed among the instruments of the set. A set of instruments soconfigured will be referred to as the critical path set.

FIG. 10C shows the cross section 56, which includes seven circles, i.e.,circles 1003 1009. The circles can define six annuli of equal area. Thecircles 1003 1009 have, respectively, diameters d₀-d₆ and radii r₀-r₆.Note that circle 1003 and 1009 correspond, respectively, to theperimeters of cross section 50 and cross section 56. Each annulus can bedefined by its outer and inner circle and, furthermore, can representthe amount of dental material that an instrument is used to remove. Theinstruments can be sized so that their cross sections at D₆ correspondto the circles 1004-1009. That is, the nth instrument is to have a D₆cross section diameter of dn. The first instrument, for example, is tohave a D₆ cross section diameter of d₁.

As can be seen from FIG. 10C, an annulus has an outer circle and innercircle. The area of an annulus can be calculated by taking thedifference between the area of the outer circle and the area of theinner circle. Given that ECS is to be enlarged sequentially from 0.20 to0.70 mm at D₆ (and 0.20 to 1.2 mm at D₁₂), the area of the annulusdefined by circles 1003 and 1009, which as discussed can represent thetotal amount of material to be removed at or around D₆, can becalculated asA _(outer circle) −A _(inner circle)=π(0.35)²−π(0.1)²A of annulus at D ₆=π(0.1225)−π(0.01)A of annulus at D ₆=π(0.1125)

For a set of six instruments, the amount of material allotted to eachinstrument to remove can be represented by diving the above calculatedtotal area by six. Each instrument would then have an increase in crosssectional area at D₆ by one-sixth the area calculated above (i.e., thearea of the annulus defined by circles 1003 and 1009 at D₆), which iscalculated as:(A of annulus at D ₆)/6=[π(0.1125)]/6=π0.01875

In general, a formula to determine the D6 radius of an nth instrumentcan be defined asπr _(n) ²=π(r _(n-1))²+π0.01875r _(n) ²=(r _(n-1))²+0.01875r _(n)=[(r _(n-1))²+0.01875]^(1/2)  [1]where n represents the sequential position of the instrument in thesequence of instruments in the set, the sequence being based on size.The first instrument in the set, for example, is usually the smallestsized instrument and is the one of the set that is first used to preparethe ECS. Thus, n is an integer that is at least 1. Additionally, r_(n)is the radius of the D₆ cross section of the nth instrument, except forr₀, which is the radius of the circle 1003 (i.e., the diameter of theunprepared ECS at D6) shown in FIG. 10C. Alternatively, the formula canbe defined asπr _(n) ²=π(r ₀)² +n(π0.01875)r _(n) ²=(r ₀)² +n(0.01875)r _(n)=[(r ₀)² +n(0.01875)]^(1/2)  [2]

Furthermore, the radius of the D₆ cross section of smallest instrumentof the set, i.e., instrument No. 1, which corresponds to the radius ofthe ECS cross section resulting after the first instrument is used toprepare the ECS isr ₁ ² =ro ²+0.01875r ₁ ²=(0.1)²+0.01875r ₁ ²=0.010+0.01875r ₁ ²=0.0388r ₁=0.1696

Thus, the diameter of the D₆ cross section of the first instrument isd ₁=0.339 mm

Using either formula [1] or formula [2], the values for d₂, d₃, d₄, d₅,and d₆ at D₆ can be derived. Table 1 shows the derived cross-sectionaldiameters, d₁, d₂, d₃, d₄, d₅, and d₆, that occur at D₀, D₆, and D₁₂.The superscript percentages, e.g., those listed in row D₆, eachrepresents the percentage increase in cross-sectional area from theprevious diameter or instrument. The average of these changes from d₁-d₆at D₆ is 24%. The average of the changes from d₁-d₅ at D₆ is 29%.

TABLE 1 d₀ d₁ d₂ d₃ d₄ d₅ d₆ D0 0.20 0.20 0.20 0.20 0.20 0.20 0.20 D60.20 0.339^(70%) 0.436^(28%) 0.515^(15%) 0.583^(13%) 0.644^(10%)0.7000^(8%) D12 0.20 0.522 0.712 0.86 0.986 1.098 1.2

Table 2 show the cross section diameters of a set of instruments that issimilar to the critical set. The average increase in diameter is 29%,which, notably, is achieved by logarithmic progression and not linearprogression of conventional set, as is the case in conventional sets ofinstruments.

TABLE 2 15/05 20/06 24/07 27/08 29/09 30/10 D1 15 20 24 27 29 30 D2 2026 31 35 38 40 D3 25 32 38 43 47 50 D4 30 38 45 51 56 60 D5 35 44 52 5965 70 D6   40^(100%)   50^(25%)   59^(18%)   67^(13%)   74^(10%) 80^(8%) D6      .339^(70%)      .436^(28%)      .515^(15%)     .583^(13%)      .644^(10%)      .7000^(8%) Critical Set D7 45 56 6675 81 90 D8 50 62 73 83 90 100  D9 55 68 80 91 99 110  D10 60 74 87 99108  120  D11 65 80 94 107  117  130  D12 70 86 101  115  126  140 

Table 3 shows the cross section diameters of another set of instrumentsthat is similar to the critical set. The percent change in the D6 crosssection diameter of this set is also 29%. Note however, that there is asubstantial increase in the diameter of the instruments from d5 to d6,which can still be considered safe, because the rigidity of thisinstrument prevents any meaningful flexure eliminating fatigue andbreakage.

TABLE 3 15/04 20/05 24/06 27/07 30/08 35/10 D1 15 20 24 27 30 35 D2 1925 30 34 38 45 D3 23 30 36 41 46 55 D4 27 35 42 48 54 65 D5 31 40 48 5562 75 D6   35^(75%)   45^(28%)   54^(20%)   62^(15%)   70^(13%)  85^(21%) D6      .339^(70%)      .436^(28%)      .515^(15%)     .583^(13%)      .644^(10%)      .7000^(8%) Critical Set

Table 4 shows the cross section diameters of yet another set ofinstruments that is similar to the critical set. These instrumentscorresponds very closely to a critical set at D6. The average increasein the D6 cross section diameter from the first to the fifth instrumentis 29%, which is identical to the critical set. The instrument with thesmallest diameter, the 15/04 can also be discounted or set aside from acritical set, because it is extremely flexible and intrinsically safe.

Optionally, where a set includes one or more instruments that have crosssectional diameters that vary along the shaft, as is the case for theinstruments represented in the tables included in this specification,one can roll the cuttings edges of portions of an instrument that hasthe same diameter as another instrument of the set. Doing so can reducethe resistance to which the instrument is subject and make theinstrument even less predisposed to failure. In the tables, shaded cellsindicate that the portions listed in the cells can have rolled edges.

TABLE 4

Table 5 shows the cross section diameters of yet another set ofinstruments that is similar to the critical set. These instrumentsdeviate slightly from the critical set; however, it mimics tips sizesand tapers that are familiar to most practitioners.

TABLE 5 15/04 20/05 25/06 30/07 35/08 40/10 D1 15 20 25 34 36 40 D2 1925 31 41 44 50 D3 23 30 37 48 52 60 D4 27 35 43 55 60 70 D5 31 40 49 6268 80 D6   35^(75%)   45^(28%)   55^(22%)   69^(25%)   76^(11%)  90^(18%) D6      .339^(70%)      .436^(28%)      .515^(15%)     .583^(13%)      .644^(10%)      .7000^(8%) Critical Set

These calculations and the instrument sets that have been proposed areagain, functions of the formula for the area of a circle. Similarcalculations can be also be made using formulas for circumference of acircle or the surface area of a frustum. Further, calculations can alsobe made using the average of the surface areas of all the intersectingdiameters of the critical path, their circumferences, the surface areasof the frustums or portions or combinations thereof.

One alternative, for example, can be implemented such that the abovedescribed equal distribution of dental material to be removed can becalculated not only for the D₆ cross section, but for any combination ofD₁-D₆ cross sections, or even for any cross section or every crosssections along the axis of the shaft.

The above described differences in D₆ cross section areas ofinstruments, for example, the difference in the D₆ cross section areasof the first and second instruments and the difference in the D₆ crosssection areas of the second and third instruments, need not be exactlyequal. The differences in areas can be substantially equal, for example,one difference being no more than twice the other difference.

The cross sections used in the above described model need not be D₆cross sections but can be cross sections substantially the same as theD₆ cross section. Cross sections within 5 millimeters of the D₆ crosssection, for example, can be used in the model.

As an another alternative, a critical set can be defined based on acalculation of the effective contact area of each endodontic instrumentin a set of endodontic instruments. The effective contact area is theexposed surface area of a cone like shape, which, for an endodonticinstrument, is defined as shown in FIGS. 11A and 11B. The exampleendodontic instrument 1102 shown includes a working portion 1104, whichas discussed above, is the portion of the instrument that includescutting edges, e.g., cutting edge 1106. The working portion includescross sections, each defining a circle into which the correspondingcross section can be inscribed. The circle having the greatest diameteris usually defined by the largest cross section, which usually occurs ator near the proximal end of the working portion (i.e., the end that isclosest to the shank and farthest away from the tip of the endodonticinstrument). The circle having the smallest diameter is usually definedby the smallest cross section, which usually occurs at or near thedistal end of the working portion (i.e., the end that is closest to thetip). In most cases, the described largest and smallest circles are thebase 1108 and the top 1110, respectively, of a cone like shape 1112. Thebase 1108 has a first diameter and the top 1110 has a second diameterthat is usually smaller than the first diameter. The distance along thelongitudinal axis 1114 of the endodontic instrument defines the height1116 of the cone like shape 1112. This cone like shape 1112 has anexposed surface area called a frustrum, which excludes the surface areaof the base 1108 and the top 1110. The exposed surface area can bemathematically defined, for example, by integrating the circlesdescribed along the height of the cone like shape.

When a critical set is based on the described effective contact area(i.e., the described surface area of the cone like shape), theendodontic instrument in a critical set of endodontic instruments can besized such that the increase of the effective contact area from oneendodontic to the next greater sized endodontic instrument issubstantially the same. FIGS. 11C and 11D show six example cone likeshapes of an example critical set of endodontic instruments. The changein surface area between any adjacent shapes is substantially constant.In cases where an endodontic instrument includes a working portion thatruns to the tip of the instrument, the cone like shape is a cone.

Alternatively, a critical set can be defined differently from the waysdescribed above. The criterion that should be satisfied is that eachendodontic instrument in a set be subject to substantially the samelevel of torsional stress. Distributing the torsional stress asdescribed reduces breakage.

Distribution of the torsional stress can be fined tuned by changing theworking portion of an instrument, as was described above. Reducing theworking portion of an instrument, for example, can reduce the level oftorsional stress to which the instrument is subject during its use. FIG.12 shows an implementation of a set of endodontic instruments where theworking portion varies from instrument to instrument so that theinstruments are subject to substantially the same level of stress duringtheir use. (The dashed lines 1202 and 1204 delimit the workingportions.)

A critical set can include more or less than six instruments. The numberof instrument depends on the amount of dental material to be removed,with more instruments being needed to removed more dental material. Thenumber of instruments, however, should not be so numerous as to impededthe ECS preparation by requiring the practitioner to frequently switchinstruments.

Critical Path Set that Includes a Combination of Reversed HelixInstruments and Non Reversed Helix Instruments

As discussed above, using a reversed helix instrument can reduce therisk of the instrument binding with the tooth being worked. Such aninstrument can thus be extremely useful during the initial removal ofdentin (a preliminary enlargement phase of ECP), when the risk ofbinding can be great. However, during later stages of ECP, the haulingcapacity of the reversed helix instrument can be reduced as dentin chipsaccumulate in the flutes of the instrument. The accumulation can preventthe reversed helix instrument from cutting into the tooth being work tothe instrument's full working length. Moreover, the reversed helixinstrument can includes flutes that have steep helix angles (andconsequently large pitch) and/or deep flutes. When it includes thesefeatures, the reverser helix instrument can leave an irregular surfacearea on the walls of a canal that require additional planning.

In one implementation, a set of critical path instruments, for example,the set listed in Table 3 above, includes a combination of instrumentsthat have reversed helices and instruments that do not have reversedhelices. The first, third, and fifth instruments (i.e., sizes 15/4,24/06, and 30/08) of the set each has reversed helices. The second,fourth, and sixth instruments (i.e., sizes 20/05, 27/07, and 35/10) ofthe set each does not have reversed helices. One would thus use, in analternating fashion, a reversed helix instrument and a non reversedhelix instrument during a preparation stage from D1 to D6. In a sense,the reverse helix instrument can be used as a rough reamer and the nonreversed helix instrument subsequent to the reversed helix instrument inthe set would be used as a finishing reamer.

Because the described accumulation does not occur with instrumentswithout reversed helices, they can be used during the later stages ofECP to provided the needed hauling. Furthermore, the non reversed helixinstruments of a set can include features typical of a finishing reamer,for example, having any combination of (i) 3 or 4 leading edges, (ii)lower helix angles and shorter flute pitch, (iii) shallower flutes, (iv)less (parabolic) relief or lower relief angles behind the leading edges,(v) a neutral or negative rake angle. Instruments of a set havingreversed helix and non-reversed helix instruments, can have reduced orno radial lands, which can predispose the instrument to drag, prematurewear, and/or breakage.

As discussed above, the portion of the root canal space from D1 to D6,especially D6, is the critical or unsafe zone of the root canal space.This is area that is the narrowest, and the most curved and isassociated with the greatest number of instruments failures orfractures. In some implementations, tip sizes and tapers of instrumentsin a set, from D1 to D6, can be fabricated so that the instrumentswithout reversed helices each has a slightly smaller tip size and aslightly larger taper than does the preceding reversed helix instrument.Such tip sizes and taper allows the non reversed helix instruments of aset to readdress the apical portion of the root canal space prepared bythe preceding reversed helix instrument and capture any debris thatmight have remained, mitigating the opportunity of apical obstructionsor blockage. From D7 to D12, the safe zone of the root canal system,tapers of instruments in a set can be fabricated so that the diameter ofeach instrument of the set progressively increases.

Table 6 shows an implementation of the above described tip size andtaper configuration. Note that eight instruments have been included,four reversed helix instruments of the critical path design with lefthanded helix and right handed cut (L/R) and four non reversed helixinstruments with right handed cut and right handed helix (R/R). Notealso that the last two columns represent two different sequences withR/R. These instruments have significantly large diameters as breakage isusually not an issue.

TABLE 6 Alternate 16/04 15/05 18/06 17/07 20/08 19/09 22/10 21/12 24/10(L/R) R/R L/R R/R L/R R/R L/R R/R R/R D1 16 15 18 17 20 19 22 21 24 D220 20 24 24 28 28 32 33 34 D3 24 24 30 31 36 37 42 45 44 D4 28 29 36 3844 47 52 57 54 D5 32 34 42 45 52 56 62 69 64 D6 36 39 48 52 60 65 72 8174 D6C/S .339^(70%)* .515^(15%) .7000^(8%) D7 40 44 54 59 68 74 82 93 84D8 44 49 60 66 76 83 92 105 94 D9 48 54 66 73 84 92 102 117 104 D10 5259 72 80 92 101 112 129 124 D11 56 64 78 87 100 110 122 141 134 D12 6069 84 94 108 119 132 143 144

Optionally, the non-reversed helix instruments of a set can include allof the features of an ideal finishing reamer, for example, (i) multipleleading edges with neutral or negative rake angles (without radiallands), (ii) lower (or higher) helix angles and shorter (or longer)flute pitch, and (iii) shallow flutes. Such non reversed helixinstruments will generally remain safe when used in conjunction with thereverse helix instruments of the set. As discussed above, there may besome justification for using a non reversed helix instrument that is ofsignificantly larger diameter than the reverse helix instrumentimmediately preceding in the set. Such use can be limited depending onthe severity of the curvature of the canal being prepared.

Optionally, the reversed helix instruments of the set can have flutesthat spiral insignificantly and are substantially straight. As with thereversed helix configuration, the straight helix design also does notbind.

Swagger

As applied to an endodontic file or reamer, phenomenon of “swagger” isviewed as a transverse mechanical wave, which can be modified, and iscomparable to the transverse wave that can be produced along a stretchedrope or string. If one ties the loose end of a long rope to a stationarypoint, stretches the rope out horizontally, and then gives the end beingheld a back-and-forth transverse motion, i.e., provide an excitationforce, F_(e), the result is a wave pulse that travels along the lengthof the rope. Observation shows that the pulse travels with a definitespeed, maintaining its shape as it travels, and that the individualsegments making up the rope move back and forth in a directionperpendicular to the rope's equilibrium position. In physics, thisprinciple can be expressed mathematically by the formula y=f(x, t).Here, the equilibrium position is selected along the x-axis(corresponding to the stretched rope), and the transverse displacementof any point away from this position is y (i.e., the maximumdisplacement of the rope, or amplitude). Thus, y is a function of both x(the undisplaced position of the point) and time t. This is called thewave function.

At any time t, if one takes a picture of the instantaneous shape of therope, we observe that y varies sinusoidally with x.

This same system can function in three dimensions whereby an excitationforce is applied along both the y- and the z-axes. Here, at any time t,if one takes a picture of the instantaneous shape of the rope, we findthat y and z vary sinusoidally with x. Again, using the stretched ropeas an example, the rope is observed to produce a spiral or helical wave.Interestingly, if one applies a lateral force F_(l) somewhere along thex-axis as an excitation force, F_(e), is applied the same phenomenon isobserved; whereas, the portion of the rope distal to the lateral forcewill also spiral.

In preparing an endodontic cavity space, ECS, along the critical path,CP, with a flexible rotary instrument, a similar system can beconfigured. The instrument is bound in a somewhat fixed position at oneend, by the elbow or greater curvature of the canal, and at the otherend by the head of the hand-piece as it is rotating. Observation of anendodontic instrument with a square, or doubly symmetric cross-sectionrotating in the ECS, appears to have little, if any, deviation from theaxis of rotation. An instrument with a singly symmetric (trapezoidal ortriangular) cross section, acts like a rope with an excitation force orforces applied along the y and z-axes, or one with an excitation forcealong the y-axis and a lateral force applied somewhere along its length.The lateral force, or moment, in this instance, is the resultant that isproduced by relocating the center of mass of the instrument away fromthe x-axis.

Most of the mathematics for helical wave theory focuses onhydrodynamics, acoustics and electromagnetic fields. Fortunately,Newtonian physics provides a plethora of laws and principles, which canbe applied to this phenomenon.

If one considers the relationship between torque λ and angularacceleration α of a point mass m, one can better understand the physicalnature of swagger. Referring to FIG. 13, a mass m moving in a circle ofradius r acted on by a tangential force F_(t).

Using Newton's second law to relate F_(t) to the tangential accelerationat =rα where α is the angular accelerationF _(t) =ma _(t) =mrα.and the fact that the torque about the center of rotation due to F_(t)is λ=F_(t)≅r, one arrives atλ=mr ²α.

For a rotating rigid body made up of a collection of masses m₁, m₂ . . .m_(i) the total torque about the axis of rotation isλ=Σλ_(i)=Σ(m _(i) r _(i) ²)α.

The second equation may consider the angular acceleration of all thepoints in a rigid body as the same, so it can be taken outside thesummation. Thus, if a constant torque λ_(i), as is applied in a torquecontrolled dental hand-piece, were applied to a mass m_(i), theacceleration of m_(i) would increase or decrease exponentially as theradius r of m_(i) changed.

Torque, like the waves along a stretched rope, must be defined about aparticular set of axes. Further considering the moment of inertia, I, ofa rigid body with mass m_(i) gives a measure of the amount of resistancethe body has to changing its state of rotational motion. Mathematically,the moment of inertia isI=Σm _(i) r _(i)The expression for torque isλ=I α.

This is the rotational analogue of Newton's second law. Applying thisprincipal to the center of mass or the centroid C of the rigid bodyyieldsF _(total) =m α _(cm)where, α_(cm) is the acceleration of the center of mass.

The moment of inertia, like torque must be defined about a particularset of axes. It is different for different choices of axes. The choiceof axes is arbitrary, and may be selected in such a way which best suitsthe particular application, and its respective geometry.

Extended objects can be considered as a very large collection of muchsmaller masses glued together to which the definition of moment ofinertia given above can be applied.

Like torque, the moment of inertia depends on how the mass isdistributed about the axis. For a given total mass, the moment ofinertia is greater if more mass is farther from the axis than if thesame mass is distributed closer to the axis.

If one considers what occurs when a number of masses m₁, m₂, . . . ,m_(i) are distributed along a line, the measure of the tendency of amass m_(i) to rotate about a point at distance x away is called themoment of the body about the point. This moment is measured by thequantity mx, and takes into account that the tendency is larger wheneither the mass or the distance to the center of rotation is larger.

With more than i=2 masses (m₁, m₂, . . . , m_(i)) placed at positionsx₁, x₂, . . . , x_(i) respectively along a coordinate line, the moment Mof the system of masses is just the sum of the individual momentsm₁x₁+m₂x₂+ . . . +m_(n)i. When M=0, the system will not rotate about thepoint at the origin, and therefore is in equilibrium.

Thus, when the center of mass of the system corresponds to the axis orrotation, the system is in equilibrium and the instrument turns evenlyaround the axis. When the center of mass or the centroid or the systemis at a distance from the center of rotation, similar to an endodonticinstrument of singly symmetric cross section, the system is out ofequilibrium and will tend to swagger.

Another interpretation of moment of inertia is the capacity of the crosssection of a system to resist bending. In a two dimensional space, thismay be considered in reference to the x- and y-axes. The Second Momentof the Area can be described in the simplest terms asI _(x) =ΣA y ²

For a thin slice or cross section of a system with uniform thickness anddensity, also called a lamina, the location of the centroid along they-axis could be expressed as y=the square root of I_(x)/A.

An ideal endodontic instrument of singly symmetric cross section wouldhave a center of mass or centroid with geometric coordinates, whichallows the instrument to cut the inner and outer walls along thecritical path of the ECS evenly. This centroid would have an angularacceleration α_(cm) larger than the α_(ar) or angular acceleration atthe geometric axis or rotation with torque λ_(x) provided by the dentalhand piece. Returning to the formula for torqueλ=mr ²α

With constant torque λ, the angular acceleration α_(cm) increases withrespect to α_(ar), as r_(cm) deviates from the axis of rotation. Pointmass systems are mathematical systems, which can better describe thiseffect in two and three dimensions.

A system of point masses m₁, m₂, . . . m_(i) located in two dimensionalspace have a moment with respect to any line L in the plane; it isdefined to be the sum M_(L)=m₁d₁+m₂d₂+ . . . +m_(n)d_(i), where thed_(i) are the (directed) distances from the masses m_(i) to the line L.In particular, if the masses are located at the points with coordinates(x₁, y₁), (x₂, y₂), . . . , (x_(n)y_(n)), then the moment of the systemabout the y-axis is M_(y)=Σm_(i)x_(i) while the moment about the x-axisis M_(x)=Σm_(i)y_(i).

As a result, the formulas X=M_(y)/m, Y=Mx/m give the coordinates of apoint (X, Y) about which the system is in equilibrium in both the x- andy-directions. This point is the center of mass of the two-dimensionalsystem and the distribution of masses will balance on the plane at thispoint.

Now if instead of a discrete system of masses, consider a planar laminawhich extends across a region R in the plane, so that at each point (x,y) in R there is a variable mass density p (x, y), measured in units ofmass per unit area, then the total mass of the lamina is given by thedouble integralm=∫∫ _(R)ρ(x,y)dAThe moment M_(y) of the lamina with respect to the y-axis is then∫∫_(R)xρ(x, y) dA and similarly M_(x)=∫∫_(R) yρ(x, y) dA. Furthermore,the same formulas used above allow one to find the center of mass (X, Y)of the lamina.

If the lamina has uniform density, that is, if ρ (x, y)=ρ is constant,then the computation of the center of mass simplifies to

$\begin{matrix}{\left( {X,Y} \right) = \left( {{\int{\int_{R}{x\;{\rho\left( {x,y} \right)}{{\mathbb{d}A}/{\int{\int_{R}{{\rho\left( {x,y} \right)}{\mathbb{d}A}}}}}}}},} \right.} \\\left. {\int{\int_{R}{y\;{\rho\left( {x,y} \right)}{{\mathbb{d}A}/{\int{\int_{R}{{\rho\left( {x,y} \right)}{\mathbb{d}A}}}}}}}} \right) \\{= \left( {{\int{\int_{R}{x{{\mathbb{d}A}/{\int{\int_{R}{\mathbb{d}A}}}}}}},{\int{\int_{R}{y{{\mathbb{d}A}/{\int{\int_{R}{\mathbb{d}A}}}}}}}} \right)}\end{matrix}$Thus, the center of mass depends in this case only on the shape of thelamina and not its density. For this reason, we often refer to thecenter of mass of a lamina of constant mass density as the centroid ofthe region R that defines its shape.

In 3D-space, one can measure moments of solids about planes. Forinstance, M_(xy)=∫∫∫_(S) zρ(x, y, z) dV is the moment of the solid Sabout the xy-plane, and the center of mass (X, Y, Z) of S satisfies theequationsX=M _(yz) /m,Y=M _(xz) /m,Z=M _(xy) /m,

-   -   m=∫∫∫_(S) zρ(x, y, z) dV being the total mass of S.

In the same way that mx measures the moment of a point mass m a distancex from a point, line, or plane in 1-, 2-, or 3-space, the second momentor moment of inertia mx² measures the resistance of the body to rotationabout the point, line, or plane. In the continuous case, this leads tothe formulasI _(x)=∫∫_(R) x ²ρ(x,y)dA,I _(y)=∫∫_(R) y ²ρ(x,y)dA

for the moments of inertia for a lamina in 2-space about the coordinateaxes. BecauseI _(x) +I _(y)=∫∫_(R) x ²ρ(x,y)dA+∫∫ _(R) y ²ρ(x,y)dA=∫∫ _(R) y²ρ(x,y)dAI₀=I_(x)+I_(y) is the moment of inertia of the lamina about the origin.This applies to the mechanics of thin plates that spin about a point inthe same plane in which they lie. Moments of inertia are naturallyextended into 3D-space.

In the field of mechanical engineering called dynamics of rigid bodies,it is useful to draw free body and kinetic diagrams of a cross sectionof the body. Considering the prismatic (i.e., constant cross sectionalarea along its length) rotating instrument with a singly symmetrictrapezoidal cross section as shown in FIGS. 14 and 15. In this case theconvention for choice of axis is to orient the axis of rotation of theinstrument along the x-axis.

Referring to FIG. 14, an exemplary shaft with a trapezoidal crosssection has a centroidal axis that extends through C, which is parallelto the axis of rotation O, or the x-axis. Thus, the free body andkinetic diagrams are as shown in FIGS. 15A and 15B, respectively.ΣM _(o) =F _(t) r _(t) −Wd _(c) =Nk _(f) r _(f) −mgd _(c) =I _(o)αΣF _(n) =man=md _(c)ω²ΣF _(t) =ma _(t) =mra _(c)Where

-   -   ω=angular speed of rotation    -   N=Normal force exerted on the instrument by the root material    -   k_(f)=coefficient of friction of the material.    -   F_(f)=Force of friction=Nk_(f)    -   C=centroid of cross section, or center of mass.    -   O=center of rotation of longitudinal axis, or geometric center.    -   d_(c)=distance from center of rotation to the centroid    -   a=angular acceleration of the cross section.    -   a_(r)=relative acceleration of the mass center of the        instrument.    -   a_(c)=centripetal acceleration of the mass center of the        instrument    -   g=acceleration due to gravity=32.2 ft/s²    -   W=weight of the cross section=mg    -   r_(f)=distance from axis of rotation to application point of        F_(f).        Hence the moment equation and the vector form of the generalized        Newton's second law of motion can be written:        ΣF=ma _(r)        ΣM _(o) =Ia

When observing the rotational dynamics of a rigid body in a laboratorysetting, engineers can utilize the complete set of dynamic equationsdiscussed above to predict the motion of the body. The engineers musthave at their disposal at least as many dynamic equations as unknowns inorder to solve for the desired resultants pertaining to the particularbody in motion. Following this line of thought the engineer may applyknown geometric relationships of a particular dynamic system to deriveadditional equations, which could be useful in solving for the desiredresults.

Thus radius of gyration k_(g) of a mass about an axis for which themoment of inertia is I is defined by the equationk _(g)=(I/m)^(1/2)Thus k_(g) is a measure of the distribution of mass of a given bodyabout the axis in question and its definition is analogous to thedefinition of the radius of gyration for area moments of inertia. If allthe mass could be concentrated at a distance k_(g) from the axis thecorrect moment of inertia would be k_(g) ²m. The moment of inertia of abody about a particular axis is frequently indicated by specifying themass of the body and the radius of gyration of the body about an axis.The moment of inertia is then calculated using the equationI=k _(g) ² m

Furthermore, if the moment of inertia of a body is known about acentroidal axis it may be determined about any parallel axis. To provethis point consider the two parallel axis shown in FIG. 14, through Cand the x-axis (axis of rotation). The radial distance from the two axesto any element of mass dm are r_(o) and r_(i), and from FIGS. 15A, 15B,the separation of the axis is d_(c). Thus, if one applies a momentequation directly about the axis of rotation x_(o)ΣM _(o) =I _(o) a

From the kinetic diagram one can obtain ΣM_(o) very easily by evaluatingthe moment of the resultant about O, which becomesΣM _(o) =I _(o) a+marThis relationship is often referred to as the transfer of axis.

The acceleration components of the mass center for circular motion aremore easily expressed in normal-tangential coordinatesa _(n) =rω ² and a _(t) =ra _(r),which are the two scalar components of the force equation ΣF=ma_(r), asshown in FIGS. 16A, 16B, 16C. In applying the moment equation about themass center C it is essential to account for the moment of the forceabout the rotational axis O. Therefore by applying the transfer of axisrelation for mass moments of inertia I_(o)=I_(r)+mr_(r) ² gives usΣM _(o)=(I _(o) −mr _(r) ²)a+mr _(r) ² a=I _(o) a.

Thus one can combine the resultant force mar and resultant couple Ira bymoving the tangential component ma_(r) to a parallel position throughpoint Q, as shown in FIG. 16D. Point Q is located bymr _(r) aq=I _(r) a+mr _(r) a(r _(r))and by using the transfer of axis theorem and I_(o)=k_(o) ²m givesq=k_(o) ²/z. Point Q is called the center of percussion and has theunique property that the resultant of all forces applied to the bodymust pass through it. It follows that the sum of moments of all forcesabout the center of percussion is always zero.ΣM _(Q)=0This principal can have practical use when analyzing the case of theendodontic instrument because it gives the engineer one more equationsto utilize when solving for unknowns after measuring certaincharacteristics of its motion in the laboratory.

The swagger of an endodontic instrument can be approximated by makingcertain assumptions and by applying the stress and strain formulas forthe bending of beams derived in the field of mechanical engineeringcalled mechanics of materials. The most general case of a system offorces is one in which the forces are neither concurrent nor parallel.Such a system can always be replaced by a single resultant force actingat an arbitrary point and a resultant couple, or bending moment. Itfollows that by resolving the forces acting on the instrument at anyinstant by utilizing the dynamic equations described earlier, theresultant lateral force P and resultant couple M acting on the centroidof an arbitrary cross section can be obtained.

In this case, assume that the lateral loads act on a plane of symmetry.Since the y-axis is the plane of symmetry for an instrument with thesingly symmetric trapezoidal cross section shown in FIG. 14, it is aprincipal axis. If the material is linearly elastic, then the neutralaxis passes through the centroid C. Thus the y and z-axis are centroidalprincipal axes of the cross section. Under these conditions the normalbending stresses acting on the cross section vary linearly with thedistance from the neutral axis and are calculated from the flexureformula, σ=My/I, wherebyσ=stress=P/A

-   -   P=Lateral load acting upon the cross section    -   A=area of the cross section        To better understand this principal, consider the trapezoidal        body shown in FIG. 14. If in a laboratory setting a still        photograph is taken of an instrument having this symmetry at any        instant, a finite deflection in the segment can be observed. If        one considers that one end is relatively fixed by the dental        hand-piece, then geometric characteristics of the deflection can        be measured. In this case the instrument will flex in a similar        fashion, as would a cantilever beam subjected to a lateral load        acting transversely to the longitudinal axis.

Here it should be noted that the majority of literature published aboutthe subject of mechanics of materials was written for structuralsystems, but the same theory applies for the stress analysis of anyobject.

The load P acting a distance L from the fixed end will create internalactions or stress resultants in the form of shear forces and bendingmoments. Furthermore, after loading, the axis is bent into a curve thatis known as the deflection curve of the beam.

For reference purposes, is constructed a system of coordinate axes withits origin at the support. The positive x-axis is directed along thelongitudinal axis (i.e., across the page from left to right) and they-axis is positive downward. The z-axis is directed inward (that is,away from the reader and into the page) so that the axes form aright-handed coordinate system.

The mechanical properties of materials used in most dental tools such asa flexible rotary instrument are determined by laboratory testsperformed on small specimens of the material. After performing a tensionor compression test of the material at various magnitudes of the load, adiagram of stress vs. strain can be prepared. Such a diagram is shown inFIG. 17.

Most structural materials have an initial region in the stress straindiagram in which the material behaves both elastically and linearly. Theregion is important in engineering because many machines are designed tofunction at low levels of stress in order to avoid permanent deformationfrom yielding. Linear elasticity is a property of many solid materialssuch as alloys of metal as is the case with nickel titanium from whichmany flexible endodontic instruments are constructed.

The linear relationship between stress and strain can be expressed bythe equationσ=Eεin which c is the strain and E is a constant of proportionality known asthe modulus of elasticity. The modulus of elasticity is the slope of thestress strain diagram in the linearly elastic region and its valuedepends on the material being used. This relationship is known asHooke's Law, named for the famous English scientist Robert Hooke(1635-1703). Hooke measured the stretching of long wires supportingweights and observed that the elongations were linearly proportional tothe respective loads applied by each weight.

To obtain the general equations of the deflection curve for a prismaticbeam (i.e. constant cross-sectional area along the length of the beam),consider the cantilever beam shown in FIG. 18A. One can place the originof coordinates at the fixed end. Thus, the x-axis of the beam isdirected positive to the right and the y-axis is directed positivedownward. Here, assume that the xy plane is the plane of symmetry andthe all loads act in this plane. It follows that the xy plane is theplane of bending. The deflection of the beam at any point m₁ at distancex₁ from the origin is the translation, or displacement of that point inthe y-direction, measured form the x-axis to the deflection curve. Whenv is expressed as a function of x we have the equation of the curve.

The angle of rotation of the axis of the beam at any point m₁ is theangle between the x-axis and the tangent to the deflection curve. Theangle is positive when clockwise, provided the x and y axes have thedirections selected.

Now consider a second point m at distance ds further along thedeflection curve at distance x+dx from the origin. The deflection atthis point is v+dv, where dv is the increment in deflection as we movefrom points m₁ to m₂. Also, the angle of rotation at m₂ is θ+dθ, wheredθ is the increment in angle of rotation. At points m₁ and m₂, we canconstruct lines normal to the tangents to the deflection curve. Theintersection of these normals locates the center of rotation p. FromFIG. 18A we see that ρdθ=dx. Hence, the curvature k (equal to thereciprocal of the radius of curvature) is given by the followingequationk=1/ρ=dθ/dsThe sign convention selected for curvature is that the angle θ increasesas we move along the beam in the positive x direction.

The slope of the deflection curve is the first derivative dv/dx, as weknow from calculus. From the above equation we see that the slope isequal to the tangent of the angle of rotation θ, because dx isinfinitesimally small, thendv/dx=tan θ or v=arctan(dv/dx)These relationships are based upon geometric considerations; thus theyapply to a beam of any material.

Most beams undergo only small rotations, or torsion when they areloaded, hence their deflection curves are very flat with extremely smallcurvatures. Under these conditions, the angle θ is a very smallquantity, thus we can make some approximations that simplify our work.This relationship holds true for the flexible rotary instrument becausewhile the deflection curve is helical due to the rotation of the centerof mass around the center of rotation, endodontic instruments areconstructed with a material of such rigidity, as compared to the rootmaterial, that there is no binding within the root cavity. Hence thetorsion about the axis of rotation, or the x-axis, is very small.

As can be seen from FIG. 18Bds=dx/cos θand since cos θ≅1, when θ is small, we obtainds≅dxTherefore, the curvature becomesk=1/ρ=dθ/dxThus for small torsional forces in the beam, the angle of rotation andthe slope are equal. By taking the derivative of θ with respect to x, wegetk=1/ρ=dθ/dx=d ² v/dx ²This formula relates the curvature k to the deflection v of the beam,and is valid for a beam of any material. If the material is linearlyelastic, and follows Hooke's law, the curvature isk=1/ρ=−M/EIin which M is the bending moment, and EI is the flexural rigidity of thebeam. It follows thatdθ/dx=d ² v/dx ² =−M/EIwhich is the differential equation of the deflection curve of a beam.This equation can be integrated in each particular case to find theangle of rotation θ or the deflection v.For example, by differentiating the above equation with respect to x andthen substituting the equations q=−dV]dx and V=dM/dx we obtaind ³ v/dx ³ =−V/Ed ⁴ v/dx ⁴ =q/Ewhere V is the shear force and q is the intensity of the distributedload. From this relationship and using prime notation, we can expressthe differential equations given above in the following formsEIv″=−M,EIv″′=−V and EIv″″=qFrom the derivations of these equations one can see that they are validonly when the material follows Hooke's law, and when the slopes of thedeflection curve are very small. Also it should be noted the equationswere derived considering the deflections due to pure bending anddisregarding the shear deformations. These limitations are satisfactoryfor most practical purposes. It is known from calculus that thedeflections can be obtained by integrating the shear force and loadequations, as well as by integrating the equation for bending-moment,with the choice depending on the personal preference of the designer.Note also that the deflection of the rotary endodontic instrument wouldbe comparable to the amplitude of the helical wave, which forms thedeflection curve. This statement is important because it suggests adirect means to design rotary instruments with varying magnitudes ofamplitude, thereby allowing us to “tailor” the instrument to fit aparticular width of root cavity.

The differential equations of the deflection curve of the beam describedabove are linear equations, that is, all terms containing the deflectionv and its derivatives are raised to the first power. As for a beam, theycan be applied to the dental instrument in much the same manner.Therefore, solutions of the equations for various loading conditions maybe superimposed. Thus, the deflections caused by several different loadsacting simultaneously on the instrument can be found by superimposingthe deflections caused the loads acting separately. This method forfinding deflection is known as the principal of superposition.

For instance, if v₁ represents the deflection due to a couple andresultant moment M₁ and v₂ represents the deflection caused by africtional force F_(f), then the total deflection caused by M₁ and F_(f)is v₁+v₂.

The methods described above for calculating deflections of a prismaticinstrument can also be used to find deflections of a non-prismaticinstrument. For a tapered instrument of variable symmetric or singlysymmetric cross section, the bending theory described previously for aprismatic instrument gives satisfactory results provided the angle oftaper is small.

The use of the differential equation for finding deflection is practicalonly if the number of equations to be solved is limited to one or two,and only if the integrations are easily performed. In the case of thetapered instrument, however, it may be difficult (or even impossible) tosolve the differential equation mathematically. The reason is that theexpression for the moment of inertial I as a function of length x isoften complicated and produces a differential equation with a variablecoefficient instead of constant coefficients. For the taperedinstrument, therefore, the moment of inertia is variable along itslength and the deflections cannot be found by exact mathematicalanalysis.

A close approximation, however, can be obtained which would predict thedeflection of the instrument caused by implied forces upon it by usingthe principal of superposition, and by considering a “model” of theactual instrument made up of prismatic sections, with each consecutivesection having smaller prismatic cross-sectional area. The totaldeflection of the actual tapered instrument, then, could be approximatedby superimposing the deflections of each individual section of themodel, whereby the cross sectional area of each prismatic section of themodel mimics the cross sectional areas of the actual instrument atvarious locations along its length.

Returning to the discussion of normal bending stresses and flexure, therotating endodontic instrument described earlier could experiencebending about both principal axes of the cross section. Hence we need toestablish a consistent sign convention for such moments. A segment ofthe instrument subjected to bending moments M_(x) and M_(y) acting on anarbitrarily selected cross section is shown in FIG. 19.

These moments are represented vectorially by double headed arrows andare taken as positive in the positive direction of the y and z-axis.When using the vectorial representation the right hand rule gives thesense of the moment (that is the direction of rotation of the moment).Note that the bending moments M_(y) and M_(z) act on the positive x faceof the lamina. If the bending moments act on the negative x face of thelamina then their vectors are in the negative direction of the y andx-axis.

The simplest case of asymmetric bending arises when a doubly symmetricbeam is subjected to loads acting in directions that are skew to theaxes of symmetry. If the cross section of a beam is asymmetric, theanalysis for bending becomes more complicated. Here, the case of a beamin bending is described for ease of exposition of procedures that can beapplied to the case of the rotating endodontic instrument.

Assume that a beam having the cross section shown below is subjected toa bending moment M. Consider two perpendicular axes y and z in the planeof the cross section. The task is to determine what conditions must bemet in order for these axes to be the neutral axes for bending under theaction of the moment M, as shown in FIGS. 20A, 20B, 20C, 20D.

It is reasonable to assume that the beam is bent in such a manner sothat the z-axis is the neutral axis. Then the xy plane is the plane ofbending, and the beam deflects in that plane. The curvature of the bentbeam is positive or negative according to the mathematical signconvention chosen. Thus, the normal stress acting on an element of areadA at distance y from the neutral axis isσ=−k _(y) Eythe minus sign is needed because positive curvature means that the partof the beam below the neutral axis is in compression. The force on theelement of area is σ_(x)dA, and the resultant force is the integral ofthe elemental force taken over the entire cross-sectional area. Since weare considering pure bending, the resultant force must be zero, thus,∫σ_(x) dA=−k _(y) E∫ydA=0This equation shows that the neutral axis (the z-axis) must pass throughthe centroid C of the cross section. As an alternative, we could haveassumed that the y-axis was the neutral axis, in which case the xz axisis the plane of bending.

Now consider the moment resultant of the stress Gx and assume thatbending takes place about the z-axis as the neutral axis, in which casethe stress Gx is obtained from the above equation. Then the bendingstresses about the y and z-axes areM _(z)=−∫σ_(x) ydA=−k _(z) E∫y ² dA=−kzEI _(z)M _(y)=−∫σ_(x) zdA=−k _(z) E∫yzdA=−k _(z) EI _(yz)in which I_(yz) is the product of inertia of the cross-sectional areawith respect to the y and z-axes. From these equations, certainconclusions can be drawn. If the z-axis is selected in an arbitrarydirection through the centroid, it will be the neutral axis only ifthere are moments M_(y) and M_(z) acting about both the y and z axes andonly if the moments are in the ratio established. by the equation above.However, if the z-axis is a principal axis, then I_(yz)=0 and the onlymoment acting is M_(z). In that event, we have bending in the xy planewith the moment M_(z) acting in that same plane. In other words, bendingtakes place in the same manner as for a symmetric beam. Similarconclusions can be made under the assumption that they-axis is theneutral axis.

Thus, the following important conclusion can be drawn. When anasymmetric beam is in pure bending, the plane of the bending moment isperpendicular to the neutral axis only if the y and z-axes are principalcentroidal axes of the cross section. Then, if a bending moment acts inone of the principal planes, this plane will be the plane of bending(perpendicular to the plane of bending) and the usual bending theory isvalid.

The preceding conclusion suggests a direct method for analyzing anasymmetric beam subjected to any bending moment M. One can begin bylocating the principal centroidal axes y and z. Then the applied coupleM is resolved into components M_(y) and M_(z), assumed to be positive inthe directions shown in FIGS. 20A, 20B, 20C, 20D. These components areM_(y)=M sin θ and M _(z) =M cos θwhere θ is the angle between the vector M and the z-axis. Each of thesecomponents acts in a principal plane and produces pure bending in thatsame plane. Thus, the usual stress and deflection formulas for purebending apply. The stresses and deflections obtained from M_(y) andM_(z) acting separately may be superimposed to obtain the correspondingquantities due to the original bending moment M. For instance, theresultant bending stress at any point A in the cross section isσ_(x) =M _(y) z/I _(y) −M _(z) y/I _(z)or;σ_(x)=(M sin θ)z/I _(y)−(M sin θ)y/I _(z)in which y and z are the coordinates of point A. The equation of theneutral axis nn is obtained by setting σ_(x) equal to zeroSin θ(z/I _(y))−sin θ(y/I _(z))=0The angle β between this line and the z-axis is obtained as followstan β=y/z=I _(y) /I _(z) tan βThis equation shows that in general the angles θ and β are not equal,hence the neutral axis is not perpendicular to the plane in which theapplied couple M acts.

The deflection produced by the bending couples M_(y) and M_(z) can beobtained from the usual deflection formulas. These deflections takeplace in the principal planes and can be superimposed to obtainresultant deflections, which lie in the plane that is normal to theneutral axis nn.

The preceding discussions present an arsenal of analytical tools whichcould be utilized to predict the dynamic performance of a rotatingendodontic instrument irregardless of its cross sectional geometry.These same principals can be used to predict the swagger, or bending andresultant deflections, which occur when the center of mass of theinstrument is offset away from the axis of rotation. By applying theseprocedures, the designer could tailor the instrument to cut in typicallyhard to reach areas of the canal. Such an example of a procedure is asfollows.

1. Conceptualize an instrument with a cross sectional geometry that hasan offset center of mass. The geometry in consideration should be singlysymmetric.

2. Draw free body and kinetic diagrams of the cross section, or lamina,taking into consideration the moment and frictional forces, which applyto the particular geometry.

3. Resolve the forces acting upon the lamina into a single resultantforce and resultant couple.

4. Determine the stress and bending moments of the section which resultfrom the forces applied.

5. Develop a prismatic “model” of the instrument, which mimics thegeometrical dimensions of the actual tapered instrument.

6. Calculate the deflection curve for each section of the model actedupon by the implied forces.

7. Write the differential equations, which apply to the particulardeflection curve in consideration.

8. Integrate the deferential equation along the length of each sectionto calculate its deflection. Note that this can be done for the shearand force equations, as well as for the bending moment equations,depending upon the personal preference of the designer.

9. Superimpose the deflections for each section, to obtain a closeapproximation of the deflection of the actual instrument.

10. Repeat the procedure for various cross sections until the desireddeflection is obtained (e.g. to deflect within the confines of aparticular width of root cavity).

Equilateral Swagger

Instruments fabricated from flexible materials that are sufficientlyasymmetrical (when viewed in transverse cross-section) can swagger,particularly at their tip ends. Some of these instruments include flutesthat are variable in depth and/or spaced apart unevenly (i.e., for agiven transverse cross section, the angles between the flutes are notthe same). Such instruments maintain the same transverse cross sectionalgeometry along the axis of rotation, and the spacing between the flutes,thus, does not change along the axis of rotation (although the flutesare uneven). The flutes, in such a case, are described in the instantspecification as being parallel.

Although an instrument configured as described above can swagger, theswagger is typically uneven about the axis of rotation. That is, theextent to which the instrument swings away from the axis of rotation isdifferent for different angles of rotation. In essence, the swing of theinstrument is not symmetrical about the axis of rotation.

A flexible endodontic instrument can be designed to swagger so that itsswing is symmetrical or approximately symmetrical about the axis ofrotation. That is, the swagger is equilateral or approximatelyequilateral. The swagger can be at least the same distance from the axisof rotation at two angular locations that are 180° apart. In someimplementations, equilateral swagger is a result of a superflexible bodyhaving a least portion with a center of mass located away from thebody's axis of rotation being bound at a tip end while rotating. Thatis, when the instrument is within an ECS and caused to rotate, theinstrument forms helical waves within the canal. The number of waves canvary based on the configuration of the instrument, forming 0.5, 1, 2, 3,4, 5, 6, 7, 8, 9, 10 or more sinusoidal waves within the canal.

When an instrument capable of swagger is rotated within the ECS, theportions of the instrument that swagger contact one wall of the ECS at atime. Conversely, in conventional rotary instruments, the instrument isused in such a way that all walls of the ECS are cleaned simultaneously.While instruments described herein are described as being capable ofequilaterally swaggering, it should be understood that the instrumentsare meant to attempt to equilaterally swagger. Thus, the instruments mayhave portions that approximate equilateral swagger and may not achieveperfect equilateral swagger under any or all conditions.

Equilaterial swagger can be caused by a number of different instrumentconfigurations. In some implementations, the instrument, when viewed intransverse cross-section, begins at the shank end as cross-sectionallysymmetrical and gradually becomes asymmetrical as it progresses towardthe tip end. The asymmetrical portion will swagger during use. Because aNiTi instrument is flexible and is able to change phase from martensiteto austenite when under pressure, the symmetrical portion of theinstrument may also begin to swagger. Instruments formed of othermaterials can also induce swagger in non-asymmetrical portions duringrotation. In other implementations, the instrument, when viewed intransverse cross-section, begins at the shank end as cross-sectionallysymmetrical, but becomes asymmetrical and takes on a new geometry as itmoves toward the tip end. For example, the instrument may have a squarecross-section at the shank end and a triangular cross-section at the tipend. The change in symmetry or geometry occurs in an area overlapping acutting surface.

In other implementations, the instrument has a “canted” axis of rotationor centroid. That is, the axis of rotation is not parallel to a linealong which centroids of the instrument lie. The axis of rotation andthe line along which the centroids lie can intersect and be at an angleto one another. The canted instrument can be linear, or can be curved.

Moreover, some of the configurations described below allow theinstrument to have equilateral swagger at a portion for cutting a curvedportion of the ECS, including the fulcrum (the point of greatestcurvature). The instrument can evenly cut the inner and outer wall ofthe ECS at the fulcrum. In general, an instrument can be configured tohave equilateral swagger at any point along the working portion of itsshaft. Such a feature is advantageous as the curvature of the canal canand typically does deviate in a mesial-distal or sagital plane and inthe bucco-lingual or coronal plane. In addition, the curvature of thecanal is quite variable, usually increasing slowly and then more rapidlyas one approaches the apical portion of the preparation. These changesin curvature are rarely linear.

Implementations

FIGS. 21A-21E illustrate one implementation of an endodontic instrumentdescribed herein. The endodontic instrument 110 includes three sides, istriangular in transverse cross-section, and can be utilized to removetissue and/or dentin from an ECS. The instrument 110 includes a shank1111 and a working portion 1112, which is tapered in a shank to tipdirection. The tip 1113 includes an active or cutting surface, which isconfluent with the working surface 1112 (for example, like the tip shownin FIGS. 22A-22D). Alternatively, the leading tip 1113 (of theinstrument shown in FIGS. 21A-21E) can include a non-active ornon-cutting surface, which is also confluent with the working surface1112 (for example, like the tip shown in FIG. 9C). The MxFD 1117 islocated near the shank end of the cutting surface and MnFD 1116 islocated near the tip 1113. The shank 1111 above the working portion 1112is essentially cylindrical and exhibits a slightly smaller diameter thanthe cutting surface at the MxFD. The instrument 110 includes rolled edgeportion 114, which is confluent with the working portion 1112. Thisrolled edge modification is illustrated in FIGS. 9A-9D. A fitting 1115,which is suitable for an engine driven motor with a hand-piece andchuck, or a handle utilized for manual instrumentation, is attached tothe shank 1111.

As shown in FIGS. 21A-21E, three continuous helical flutes 1120A, 1120Band 1120C are substantially concave grooves which follow thecircumference of the working surface 1112 spiraling toward the leadingtip 1113 forming concentric circles. These flutes may be equidistantfrom each other or become increasingly tighter or more numerous as theyapproach the tip. The total number of flutes from MxFD to the MnFDshould be no fewer than 16 but not greater than 24. Helical flutes1120A, 1120B and 1120 C each originate at the MxFD at separate locationsthat are equally spaced apart around the circumference of the shank 1111or more specifically at 1200 of separation. Each helical structure(i.e., the mass between the flutes) is continuous along the length ofthe cutting surface 1112 to the leading tip 1113.

With reference to FIG. 21E, it can be seen that flutes 1120A, 1120B and1120C have S shaped splines 1121A, 1121B and 1121C. The flutes 1120A,1120B and 1120C form helical cutting edges 1125A, 1125B and 1125C at theperiphery of the shank 1111. A transverse cross-section is shown of thecutting portion 1112. The helical flutes 1120A, 1120B and 1120Ccooperate to form a web or core 1126, which is essentially triangular.Areas of radial clearance or cutouts created by the flutes 1121A, 1121Band 1121C outline the web or core. These areas of clearance aredesignated by numerals 1130A, 1130B and 1130C. In transversecross-section of the shank 1111, splines 1121A, 1121B, and 1121C ofcutting flutes 1120A, 1120B, and 1120C form teardrop shaped clearanceareas of variable depth. The cutting surfaces 1125A, 1125B, and 1125C,or the perimeter of the shank, and the splines of the inner walls 1121A,1121B, and 1121C circumscribe clearance areas 1130A, 1130B, and 1130C.

With further reference to FIG. 21E, it can be seen that walls 1121A,1121B, and 1121C intersect the periphery of the shank 1111 at points1131A, 1131B, and 1131C. These intersections are equal distances apartor at 1200 of separation forming a neutral cutting angle (90° angle tothe tangent of the perimeter of shank 1111) or slightly positive rakeangle (greater than 90° to the tangent of the perimeter of the shank1111). Lines drawn connecting point 1131A, 1131B, and 1131C form anequilateral triangle. As shown in FIG. 21D, points 1131A, 1131B, and1131C intersect the periphery of the shank 1111 alternately at 110°,125°, and 125° of separation. Lines drawn connecting the point 1131A,1131B, and 1131C form an isosceles triangle. The outline of the trianglethat is formed connecting point 1131A, 1131B, and 1131C can vary. Theoutline may also be a scalene triangle with unequal sides. Thedifference in the number of degrees of separation between the longestspline and the short spline is not less than 60° and not greater than150°.

The splines 1121A, 1121B, and 1121C are S-shaped and are individuallysymmetrical. The bisector of each spline divides the spline equally intoconvex and a concave segments which form the S-shaped curve. The linesthat bisect each spline can be drawn to the center of the core 1126 andare equal in length. Further, an alternate bisector a can be drawn fromthe center of each spline through the greatest concavity the adj acentspline and perpendicular to the lines, which form the equilateraltriangle. The bisectors for each spline 1121A, 1121B, and 1121C areequal.

The greatest depth of each spline can be defined by a segment of a.These depths can vary and, furthermore, be calculated as a percentage ofthe length of a. The greatest depths of splines 1121A, 1121C, and 1121B,indicated with demarcated line segments 1137A, 1137B, and 1137C, are15%, 20%, and 25% of the length of a, respectively. The greatestconvexities of splines 1121A, 1121B, and 1121C are mirror images of thegreatest concavities of the same splines. The depth and height of eachspline can vary; however, the cross sectional diameter of the coreportion 1126 should generally not be narrower than approximately half orfifty percent of the cross sectional diameter of the shank of theinstrument.

FIGS. 23A-23D illustrate another implementation, which is four sided orrectilinear in transverse cross-section and can be utilized to removetissue and/or dentin from an ECS. The implementation shown includes ashank similar to the one described in FIGS. 21A-21E and a workingportion 1212, which is tapered in a shank to tip direction. The tip 1213can include a cutting surface, which is confluent the working surface1212 (for example, like the tip shown in FIGS. 22A-22D). Alternatively,the leading tip 1213 (of the instrument shown in FIGS. 23A-23D) caninclude a non-cutting surface, which is confluent with the workingsurface 1212 (for example, like the tip shown in FIG. 9C). The MxFD 1217is located near the shank end of the cutting surface and MnFD 1216 islocated near the tip end of the cutting surface. The shank 1211 abovethe cutting surface 1212 is essentially cylindrical exhibiting aslightly smaller diameter than the cutting surface at the MxFD (alsosimilar to the instrument described in FIGS. 21A-21E). The instrumentcan include a modified or rolled edge portion 1214, which can to beconfluent with the cutting surface 1212. This rolled edge feature isillustrated in FIGS. 9A-9D. A fitting, which is suitable for an enginedriven motor with a hand-piece and chuck or a handle utilized for manualinstrumentation, is attached to the shank at its most proximal end (alsosimilar to the fitting described in FIGS. 21A-21E).

With further reference to FIGS. 23A-23D, four continuous helical flutes1220A, 1220B, 1220C, and 1220D are substantially concave grooves whichfollow the circumference of the working surface 1212 spiraling towardthe leading tip 1213 forming concentric circles, which may beequidistant from each other or becoming increasingly tighter or morenumerous as they approach the tip 1213. The total number of flutes fromMxFD to the MnFD should be no fewer than 16 but not greater than 24.Helical flutes 1220A, 1220B, 1220C, and 1220D each originate at the MxFDat separate locations that are equally spaced apart around thecircumference of the shank 1211 or more specifically at 90° ofseparation. Each flute is continuous along the length of the cuttingsurface 1212 to the leading tip 1213.

With reference to FIG. 23D, it can be seen that flutes 1220A, 1220B,1220C, and 1220D have an S-shaped splines 1221A, 1221B, 1221C, and1221D. The flutes 1220A, 1220B, 1220C, and 1220D form helical cuttingedges 1225A, 1225B, 1225C, and 1225D at the periphery of the shank 1211.With reference to FIG. 23D, a transverse cross-section is shown of thecutting portion 1212. The helical flutes 1220A, 1220B, 1220C, and 1220Dcooperate to form a web or core 1226, which is generally square shaped.The web or core is outlined by areas of radial clearance or cut outscreated by splines 1221A, 1221B, 1221C, and 1221D. These areas ofclearance are designated by numerals 1230A, 1230B, 1230C and 1230D. Intransverse cross section of the shank, splines 1221A, 1221B, 1221C, and1221D of cutting flutes 1220A, 1220B, 1220C, and 1220D form teardropclearance areas of variable depth. Clearance areas 1230A, 1230B, 1230C,and 1230D are circumscribed by cutting edges 1225A, 1225B, 1225C, and1225D, or the perimeter of the shank, and the splines of the inner walls1221A, 1221B, 1221C, and 1221D.

With further reference to FIG. 23D, it can be seen that splines 1221A,1221B, 1221C, and 1221D intersect the periphery of the shank 1211 atpoint 1231A, 1231B, 1231C, and 1231D, respectively. These intersectionsare equal distances apart or at 90° of separation, forming a neutralangles (90° angle to the tangent of the perimeter of shank 1211) orslightly positive rake angles (greater than 90° to the tangent of theperimeter of the shank 1211). Lines drawn connecting point 1231A, 1231B,1231C, and 1231D form a square.

The splines 1221A, 1221B, 1221C, and 1221D are S-shaped and areindividually symmetrical. The bisector of each spline divides the splineequally into convex and a concave segments which form the S-shapedcurve. The lines that bisect each spline can be drawn to the center ofthe core 1226 and are equal in length. Further, an alternate bisector acan be drawn from the center of each spline through the greatestconcavity the opposite spline and is also equal in length.

The greatest depth of each spline can be defined by a segment of a.These depths can vary and, furthermore, be calculated as a percentage ofthe length of a. The greatest depths of splines 1221A and 1221C,indicated with demarcated line segments 1237A and 1237C, are 5% of thelength of a. The greatest depth of splines 1221B and 1221D, indicatedwith line segments 1237B and 1237D, are 25% of a. The greatestconvexities of splines 1221A, 1221B, 1221C, and 1221D are mirror imagesof the greatest concavities of the same splines. The depth and height ofeach spline can vary; however, the cross sectional diameter of the coreportion 1226 should generally not be narrower than approximately half orfifty percent of the cross sectional diameter of the shank of theinstrument.

FIGS. 24A-24E illustrate another implementation, which is four sided orrectagonal in transverse cross-section and can be utilized to removetissue and/or dentin from an ECS. The instrument includes a shank 311similar to the one described above with respect to FIGS. 21A-21E and aworking portion 312, which is tapered in a shank to tip direction. Thetip 313 can include a cutting surface, which is confluent the workingsurface 312 (for example, like the tip shown in FIGS. 22A-22D).Alternatively, the tip 313 can display a non-cutting surface, which isconfluent with the working portion 312 (for example, like the tip shownin FIG. 9C). The instrument includes an MxFD 317 and an MnFD 316. Theshank 311 above the above the working portion 312 is essentiallycylindrical exhibiting a slightly smaller diameter than the cuttingsurface at the MxFD. The shank here is similar to the one described inFIGS. 21A-21E. The instrument includes a modified or rolled edge portion314, which is confluent with the cutting surface 312. This rolled edgefeature is illustrated in FIGS. 9A-9D. A fitting, which is suitable foran engine driven motor with a hand-piece and chuck or a handle utilizedfor manual instrumentation, is attached to the shank at its mostproximal end also similar to the fitting described in FIGS. 21A-21E.

As shown in FIGS. 24A-24E, four continuous flutes 320A, 320B, 320C and320D are substantially concave grooves, which follow the circumferenceof the working surface 312 spiraling toward the leading tip 313 formingconcentric circles, which may be equidistant or become increasinglytighter or more numerous as they approach the tip. The total number offlutes from MxFD to the MnFD should be no fewer than 16 but not greaterthan 24. Flutes 320A, 320B, 320C, and 320D each originate at the MxFD atvarious locations spaced around the circumference of the shank, morespecifically at 80°, 1000, 80°, and 100° of separation, respectively.Each flute is continuous along the length of the cutting surface 312 tothe leading tip 313.

With reference to FIG. 24E, it can be seen that flutes 320A, 320B, 320C,and 320D have an S-shaped splines 321A, 321B, 321C, and 321D. The flutes320A, 320B, 320C, and 320D form helical cutting edges 325A, 325B, 325C,and 325D at the periphery of the shank 311. As shown in FIG. 24E, atransverse cross-section is shown of the cutting portion 312. The flutes320A, 320B, 320C, and 320D cooperate to form a web or core 326, which isessentially rectagonally shaped. The web or core is outlined by areas ofradial clearance or cut outs created by the splines 321A, 321B, 321C,and 321D. These areas of clearance are designated by numerals 330A,330B, 330C, and 330D. In transverse cross section of the shank, thesplines 321A, 321B, 321C, and 321D of flutes 320A, 320B, 320C, and 320Dform teardrop clearance areas of variable depth. Clearance areas 330A,330B, 330C, and 330D are circumscribed by cutting edges 325A, 325B,325C, and 325D, or the perimeter of the shank, and the splines 321A,321B, 321C, and 321D.

As shown in FIG. 24E, it can be seen that splines 321A, 321B, 321C, and321D intersects the periphery of the shank 311 at points 331A, 331B,331C, and 331D, respectively. These intersections are at 80°, 1000, 80°,and 100° of separation, respectively, forming neutral or slightlypositive rake angles. Lines drawn connecting point 331A, 331B, 331C, and331D form a rectangle. The difference in degrees between the longestspline and the shortest spline is 20°. Alternatively, as shown in FIG.24D, points 331A, 331B, 331C, and 331D can intersect the periphery ofthe shank at 90°, 95°, 80° and 95° of separation, respectively. Linesdrawn connecting the point 331A, 331B, 331C, and 331D also form arectangle. The outline of the trapezoid that is formed connecting point331A, 331B, 331C, and 331D can vary. The difference in the number ofdegrees of separation between the longest spline and the short splineshould not be less than 5° and not greater than 70°.

The splines 321A, 321B, 321C, and 321D are S-shaped and are individuallysymmetrical. The bisector of each spline divides the spline equally intoconvex and a concave segments which form the S-shaped curve. Alternatebisectors a and b can be drawn from the center of each spline throughthe greatest concavity the opposite spline.

The greatest depth of each spline can be defined segments of a and b.These depths can vary and, furthermore, be calculated as a percentage ofthe length of a and b. The greatest depths of splines 321A and 321C,indicated with demarcated line segments 337A and 337C, are 5% of thelength of a. The greatest depth 321B and 321D, indicated with demarcatedline segments 337B and 337D, are 5% of b. The greatest convexities ofsplines 321A, 321B, 321C, and 321D are mirror images of the greatestconcavities of the same splines. The depth and height of each spline canvary; however, the cross sectional diameter of the core portion 326should generally not be narrower than half or fifty percent of the crosssectional diameter of the shank of the instrument.

FIGS. 25A-25E illustrate another implementation, which is four sided andtrapezoidal in transverse cross-section and can be utilized to removetissue and/or dentin from an ECS. The instrument includes a shank 411similar to the one described above with respect to FIGS. 21A-21E and aworking portion 412, which is tapered in a shank to tip direction. Thetip 413 can include a cutting surface, which is confluent the workingsurface 412 (for example, like the tip shown in FIGS. 22A-22D).Alternatively, the tip 413 can display a non cutting surface, which isconfluent with the working portion 412 (for example, like the tip shownin FIG. 9C). The instrument includes an MxFD 417 and an MnFD 416. Theshank 411 above the above the working portion 412 is essentiallycylindrical exhibiting a slightly smaller diameter than the cuttingsurface at the MxFD. The shank here is similar to the one described inFIGS. 21A-21E. The instrument includes a modified or rolled edge portion414, which is confluent with the cutting surface 412. This rolled edgefeature is illustrated in FIGS. 9A-9D. A fitting, which is suitable foran engine driven motor with a hand-piece and chuck or a handle utilizedfor manual instrumentation, is attached to the shank at its mostproximal end also similar to the fitting described in FIGS. 21A-21E.

As shown in FIGS. 25A-25E, four continuous flutes 420A, 420B, 420C and420D are substantially concave grooves, which follow the circumferenceof the working surface 412 spiraling toward the leading tip 413 formingconcentric circles, which may be equidistant or become increasinglytighter or more numerous as they approach the tip. The total number offlutes from MxFD to the MnFD should be no fewer than 16 but not greaterthan 24. Flutes 420A, 420B, 40C, and 420D each originate at the MxFD atvarious locations spaced around the circumference of the shank, morespecifically at 90°, 100°, 70°, and 100° of separation, respectively.Each flute is continuous along the length of the cutting surface 412 tothe leading tip 413.

With reference to FIG. 25E, it can be seen that flutes 420A, 420B, 420C,and 420D have an S-shaped splines 421A, 421B, 421C, and 421D. The flutes420A, 420B, 420C, and 420D form helical cutting edges 425A, 425B, 425C,and 425D at the periphery of the shank 411. As shown in FIG. 25E, atransverse cross-section is shown of the cutting portion 412. The flutes420A, 420B, 420C, and 420D cooperate to form a web or core 426, which isessentially rectagonally shaped. The web or core is outlined by areas ofradial clearance or cut outs created by the splines 421A, 421B, 421C,and 421D. These areas of clearance are designated by numerals 430A,430B, 430C, and 430D. In transverse cross section of the shank, thesplines 421A, 421B, 421C, and 421D of flutes 420A, 420B, 420C, and 420Dform teardrop clearance areas of variable depth. Clearance areas 430A,430B, 430C, and 430D are circumscribed by cutting edges 425A, 425B,425C, and 425D, or the perimeter of the shank, and the splines 421A,421B, 421C, and 421D.

As shown in FIG. 25E, it can be seen that splines 421A, 421B, 421C, and421D intersect the periphery of the shank 411 at points 431A, 431B,431C, and 431D, respectively. These intersections are at 90°, 100°, 70°,and 100° of separation, respectively, forming neutral or slightlypositive rake angles. Lines drawn connecting points 331A, 331B, 331C,and 331D form a trapezoid. The difference in degrees between the longestspline and the shortest spline is 30°. Alternatively, as shown in FIG.24E, points 331A, 331B, 331C, and 331D can intersect the periphery ofthe shank 311 at 80°, 100°, 80°, and 100° of separation, respectively.Lines drawn connecting the points 431A, 431B, 431C, and 431D also form atrapezoid. The outline of the trapezoid that is formed connecting points431A, 431B, 431C, and 431D can vary. The difference in the number ofdegrees of separation between the longest spline and the short splineshould not be less than 5° and not greater than 70°.

The splines 421A, 421B, 421C, and 421D are S-shaped and are individuallysymmetrical. The bisector of each spline divides the spline equally intoconvex and a concave segments which form the S-shaped curve. Alternatebisectors a, b, and c can be drawn from the center of each splinethrough the greatest concavity the opposite spline.

The greatest depth of each spline can be defined as segments of linesIVa, IVb, and IVc. These depths can vary and, furthermore, be calculatedas a percentage of the length of lines IVa, IVb, and IVc. The greatestdepth of spline 421A, indicated with demarcated line segments 437A, is5% of the length of IVa. The greatest depths of splines 421B, 421C, and421D can be similarly indicated by demarcated line segments 437B, 437C,and 437D (437D is not shown but is the same length as 437B). Thegreatest convexities of splines 421A, 421B, 421C, and 421D are mirrorimages of the greatest concavities of the same splines. The depth andheight of each spline can vary; however, the cross sectional diameter ofthe core portion 426 should generally not be narrower than half or fiftypercent of the cross sectional diameter of the shank of the instrument.

FIG. 26 provides a diagram of the part of an instrument 1802 that swingsaway from the axis of rotation different distances, depending on theangle of rotation, and thus swaggers in an un-equilateral manner. Theextent to which the instrument 1802 swings away from the axis ofrotation 1804 at a first angle of rotation, 90 degrees, is much greaterthan the extent to which the instrument 1802 swings away from the axisof rotation 1804 at a second angle of rotation, 270 degrees. In essence,the swing of the instrument is not symmetrical about the axis ofrotation.

Referring to FIG. 27, a schematic of an instrument 2001 is shown that iscapable of equilateral swagger. In one implementation, the profile ofthe instrument when viewed in transverse cross-section begins in itsmost proximal segment (near the shank end) as a completely symmetricalcross section. Gradually, the cross section becomes asymmetrical as itprogresses distally (toward the tip end). A longitudinal view from theproximal end 2016 to the distal end 2036 would exhibit the symmetrychange. At the proximal end 2016, a cross section of the instrument 2001has points 2018 that are equidistant from a center of the instrument2001 and overlap with an equilateral triangle 2020 that fits within acircle 2022 that would be formed by the points 2018 as the instrument2001 rotates. At the distal end 2036, the cross section of theinstrument 2001 has points 2032 that overlap with an isosceles triangle2030 that fits within a circle 2038 formed by the points 2032 as theinstrument 2001 rotates. The described change from a symmetrical to anasymmetrical cross-section can be gradual or progressive, but a rapid orprogressive change is viewed to be more efficient. A gradual change canbe a change that is similar over each segment, such as when the changebetween D₂ and D₃ is equal to the amount of change from D₃ to D₄ andfrom D₄ to D₅. A more progressive change may be that the amount ofchange changes geometrically or exponentially from the segment betweenD₄ and D₅ and the segment between D₃ and D₄. The change from symmetricalto asymmetrical can mean that the some of the flutes become closertogether while others move further apart, one or more lands may extendfurther from the instrument than other of the lands, or geometry canchange, such as when a triangular cross section with sides having thesame length becomes a triangular cross section where at least one sideis longer than another, or when a square cross section becomestrapezoidal.

Yet another instrument that is capable of equilateral swagger can changeboth its symmetry and geometry along the length. The profile of theinstrument when viewed in transverse cross-section begins in its mostproximal segment as symmetrical, but takes on asymmetry and new geometrysimultaneously. Referring to FIGS. 28, 28A and 28B, in oneimplementation, the cross section of the instrument 2101 begins, forexample, as four-sided or rectangular and is symmetrical at a shank end2112 and becomes three sided and asymmetric at a tip end 2131. Otherexamples are also possible, such as a three-sided polygon changing intoa four, five or six-sided polygon, a four-sided polygon changing to afive or six-sided polygon, a five-sided polygon changing to a three,four, or six-sided polygon or a six-sided polygon changing to a three,four or five-sided polygon. The term polygon approximates the shape andis not meant to indicate that the sides necessarily are linear.

Referring to FIGS. 29, 29A, 29B, 29C, and 29D, a schematic of aninstrument 2201 features a profile that has a canted longitudinal axis.That is, the centroids 2210 of the multiple cross sections of the cantedportion 2242 of the instrument 2201 do not lie on the axis of rotation2216. The instrument 2201 here can include a portion 2222 that has atleast one cross-section (for example, near the shank end) that has acentroid 2210 that is on the axis of rotation 2216 and another portion2242 that has at least one cross section (for example, near the tip end)that has a centroid that is off the axis of rotation 2216. The geometrycan change or stay the same down the length of the instrument. Thecanted instrument can have a linear section, where at least the portionof the instrument having the working surfaces is linear, see instrument2250 as shown in FIG. 29C, or the entire instrument can be linear, seeinstrument 2260 as shown in FIG. 29D.

Referring to FIGS. 30A and 30B, in some implementations, a portion ofthe canted instrument 2301, 2301′ is curved. An instrument with avariable centroid, can have a gradual or progressive curve in theprofile itself. That is, the centroids of the cross sections define acurve. The tip 2315, 2315′ of the instrument 2301, 2301′ can lie on oroff of the axis of rotation 2320, respectively.

Any combination of the above described design features can be employedto create a hybrid design, which can be an enhancement of the basicdesigned or employed for a special case that may require customizedinstruments. For example, the working surfaces described in FIGS. 21-25can be used with the instruments described in FIGS. 27-30B.

The changes in cross section geometry is not limited to those describedabove. The instrument can have, for example, a change in cross sectiongeometry such that the cross sections of the instrument portion forcutting at the curved portion of the ECS are asymmetrical while thecross sections of the tip and end portions are symmetrical. Moreover,the change need not be gradual and can be accelerated, for example, atthe instrument portion for cutting at the curved portion of the ECS.

Referring to FIG. 31A, 31B, as described herein, portions of theendodontic instrument capable of equilateral swagger will bend away fromthe axis of rotation when being used, such as with a motorized drill. Insome implementations, the swaggering instrument 2410 will create amechanical wave 2420, or multiples of a half of a mechanical wave. Whenviewed, the wave may appear to form helical waves that propagate up anddown within the canal. The wave spirals in the x, y and z axes and movesin three dimensions. This can allow the instrument to undulate withinthe ECS to cut away material within the ECS. The swaggering can taper asit approaches non-swaggering portions of the instrument 2410.Additionally, changes in diameter and geometry can change the amount ofswagger. FIG. 31A shows the instrument at two different locations at twodifferent points in time while the instrument rotates. FIG. 31B showsthe area covered by the instrument as it rotates. As the wavepropagates, different portions of the instrument extend from the axis ofrotation varying amounts (not shown) and may appear as a spiraling bodyto a human viewer when the instrument is rotating very fast.

Methods of Forming Instruments

The instruments described herein can be formed starting with a blankthat is then shaped to achieved a desired result. Referring to FIGS.32A, 32B, a blank 2510, such as a metal blank, can be shaped to have asubstantially uniform geometry or cross section from a shank end 2514 toa tip end 2518. The geometry is asymmetrical down the length of theblank 2510. The center of mass is the same relative to the axis ofrotation 2516 along the length of the blank 2510. As the flutes areformed, such as by milling, machining, cutting or grinding, the relativedistances between the flutes can be changed or the depths can be changedto achieve the instrument described above with reference to FIG. 27.

Referring to FIGS. 33A, 33B, a blank 2520 is shaped to have a differentgeometry at the shank end 2524 than the tip end 2528. Here, at the shankend 2524, the blank 2520 has a rectangular or square cross section. Atthe tip end 2528, the blank 2520 takes on a triangular cross section.Between the triangular cross section and the rectangular cross section,the blank takes on a trapezoidal cross section. The blank 2520 has anaxis of rotation 2526. The desired flute pattern can then be formed inthe instrument. Such an instrument is described above with respect toFIG. 28.

Referring to FIGS. 34A, 34B, a blank 2530 is cut into a canted shape. Atthe shank end 2534, the blank 2530 has a center of mass or geometriccenter that is on a first axis 2540. Toward the tip end 2538, the blank2530 has a center of mass or geometric center that is on a second axis2542. The first access 2540 is parallel to the second access 2542, butthe two do not overlap. If the instrument were cut perpendicular toeither of the axes, the cross section of the instrument would besymmetrical. In some implementations, the blank is formed with aflexible metal, such as a memory metal and the shape is achieved bymachining the blank 2530, rather than bending the blank. The instrumentis then machined to form the desired cutting surfaces. Such a cantedinstrument formed from blank 2530 is described in FIG. 29D. In someimplementations, the blank is symmetrical from the shank end 2534 to thetip end 2538.

Referring to FIGS. 35A, 35B, a blank 2550 is cut into a curved shape toform a curved instrument in a similar manner as described with respectto FIGS. 34A and 34B. The blank 2550 curves away from an axis 2552 thatintersects a center of the blank 2550 at the shank end 2554.

An instrument that is bent into the desired configuration and thenannealed (such as is often done with conventional dental tools,regardless of their geometry) may not exhibit the same type ofswaggering, i.e., equilateral swaggering, as a similarly shapedinstrument that has been shaped by cutting or machining.

Alternatives

A number of implementations of the invention have been described.Nevertheless, it will be understood that various modifications may bemade without departing from the spirit and scope of the invention. Forexample, in other implementations, similar instruments can include 5 or6 flutes. The shanks and/or metal blanks from which these instrumentscan be fabricated and have slightly larger diameters providing enoughmaterial to facilitate the increased number of flutes. The flutes,therefore, would require fewer spirals per unit length. Instruments ofincreasing size, or diameter, become increasingly less flexible.Implementing more flutes and/or cutting the flutes deeper into the metalblanks during manufacture can facilitate compensation for the decreasein flexibility. In addition, wider and deeper spaces also providegreater opportunity to haul out debris from the apex to the coronalaspect of the tooth. Flexible materials other than Ni—Ti can be used toform the instrument. Also, although files and reamers are described inthis document, an instrument having the features described herein can beapplied to either type of device. Accordingly, other embodiments arecontemplated.

What is claimed is:
 1. An endodontic instrument for preparing anendodontic cavity space, the instrument comprising: a shank configuredfor attachment to a motor to rotate the instrument about an axis ofrotation; and a body extending from the shank, the body having a shankend where the body extends from the shank and a tip end opposite of theshank end, the body having a working surface between the shank end andthe tip end, the body having a single constant taper from the shank endto the tip end such that the tip end has a diameter that is less than adiameter of the shank end, the working surface including a plurality ofhelices, each helix of the plurality of helices including a helixcutting edge, each helix cutting edge spiraling around the body in afirst direction of rotation along the working surface, each helixcutting edge having one or more cross cuts therein, the one or morecross cuts spiraling around the body in a second direction of rotationalong the working surface, the second direction of rotation differingfrom the first direction of rotation, wherein the body comprises aplurality of transverse cross-sections, each transverse cross-section ofthe body having a center of mass, the body having a center of mass pathdefined by the centers of mass of the plurality of transversecross-sections of the body, wherein at least a portion of the center ofmass path between the tip end and the shank end spirals around the axisof rotation along a length of the axis of rotation.
 2. The instrument ofclaim 1, wherein the one or more cross cuts include cross cut cuttingedges that spiral around the body in the second direction of rotationalong the working surface.
 3. The instrument of claim 1, wherein theplurality of helices and the one or more cross cuts are at right anglesto each other.
 4. The instrument of claim 1, wherein the plurality ofhelices and the one or more cross cuts are not at right angles to eachother.
 5. The instrument of claim 1, wherein the plurality of helicescomprises four helices.
 6. The instrument of claim 1, wherein the one ormore cross cuts are V-shaped.
 7. The instrument of claim 1, wherein atleast one transverse cross-section of the working surface has a centerof mass that is offset from the axis of rotation.
 8. The instrument ofclaim 1, wherein the plurality of helices are reversed helices.
 9. Theinstrument of claim 1, wherein a portion of the working surface does nothave cutting edges.
 10. The instrument of claim 1, wherein the one ormore cross cuts have semi-circular cross-sectional shapes.
 11. Theinstrument of claim 1, wherein a portion of the body is curved.
 12. Theinstrument of claim 1, wherein the tip end is offset from the axis ofrotation.
 13. The instrument of claim 1, wherein the tip end coincideswith the axis of rotation.
 14. The instrument of claim 1, wherein atleast one transverse cross-section of the body is polygonal.
 15. Theinstrument of claim 1, wherein the body comprises nickel-titanium. 16.The instrument of claim 1, wherein the one or more cross cuts areU-shaped.
 17. The instrument of claim 1, wherein each helix of theplurality of helices defines a respective flute, wherein at a firsttransverse cross-section of the body the respective flutes have adifferent depth than at a second transverse cross-section of the body,and wherein a relative distance between the respective flutes isdifferent at the first transverse cross-section in comparison to thesecond transverse cross-section.
 18. A method of preparing an endodonticcavity space, the method comprising: inserting an instrument into theendodontic cavity space, wherein the instrument comprises: a shankconfigured for attachment to a motor to rotate the instrument about anaxis of rotation; and a body extending from the shank, the body having ashank end where the body extends from the shank and a tip end oppositeof the shank end, the body having a working surface between the shankend and the tip end, the body having a single constant taper from theshank end to the tip end such that a diameter of the tip end is smallerthan a diameter of the shank end, the working surface including aplurality of helices, each helix of the plurality of helices including ahelix cutting edge, each helix cutting edge spiraling around the body ina first direction of rotation along the working surface, each helixcutting edge having one or more cross cuts therein, the one or morecross cuts spiraling around the body in a second direction of rotationalong the working surface; the second direction of rotation differingfrom the first direction of rotation wherein the body comprises aplurality of transverse cross-sections; each transverse cross-section ofthe body having a center of mass, the body having a center of mass pathdefined by the centers of mass of the plurality of transversecross-sections of the body; wherein at least a portion of the center ofmass path between the tip end and the shank end spirals around the axisof rotation along a length of the axis of rotation; and rotating theinstrument about the axis of rotation while the instrument is in theendodontic cavity space.
 19. The method of claim 18, wherein the one ormore cross cuts include cross cut cutting edges that spiral around thebody in the second direction of rotation along the working surface. 20.The method of claim 19, wherein, during said method of preparing theendodontic cavity space, cutting is performed by the helix cutting edgesand by the cross cut cutting edges.
 21. The method of claim 18, whereinthe body forms helical waves during said rotating the instrument, andwherein the helical waves are formed because a transverse cross-sectionof the body has a center of mass that is spaced apart from the axis ofrotation.
 22. The method of claim 18, wherein, during said rotating theinstrument, one or more helix cutting edge of the plurality of helicesare out of contact with a wall of the endodontic cavity space.